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The underlying goal of this work is to reduce the uncertainty related to thermally induced stress prediction. This is accomplished by considering use of non-linear material behavior, notably path dependent thermal hysteresis behavior in the elastic properties.
Primary novel factors of this work center on two aspects.
1. Broad material characterization and mechanistic material understanding, giving insight into why this class of material behaves in characteristic manners.
2. Development and implementation of a thermal hysteresis material model and its use to determine impact on overall macroscopic stress predictions.
Results highlight microcracking evolution and behavior as the dominant mechanism for material property complexity in this class of materials. Additionally, it was found that for the cases studied, thermal hysteresis behavior impacts relevant peak stress predictions of a heavy-duty diesel particulate filter undergoing a drop-to-idle regeneration by less than ~15% for all conditions tested. It is also found that path independent heating curves may be utilized for a linear solution assumption to simplify analysis.
This work brings forth a newly conceived concept of a 3 state, 4 path, thermally induced microcrack evolution process; demonstrates experimental behavior that is consistent with the proposed mechanisms, develops a mathematical framework that describes the process and quantifies the impact in a real world application space.

Turbomachinery plays an important role in many cases of energy generation or conversion. Therefore, turbomachinery is a promising approaching point for optimization in order to increase the efficiency of energy use. In recent years, the use of automated optimization strategies in combination with numerical simulation has become increasingly popular in many fields of engineering. The complex interactions between fluid and solid mechanics encountered in turbomachines on the one hand and the high computational expense needed to calculate the performance on the other hand, have, however, prevented a widespread use of these techniques in this field of engineering. The objective of this work was the development of a strategy for efficient metamodel based optimization of centrifugal compressor impellers. In this context, the main focus is the reduction of the required numerical expense. The central idea followed in this research was the incorporation of preliminary information acquired from low-fidelity computation methods and empirical correlations into the sampling process to identify promising regions of the parameter space. This information was then used to concentrate the numerically expensive high-fidelity computations of the fluid dynamic and structure mechanic performance of the impeller in these regions while still maintaining a good coverage of the whole parameter space. The development of the optimization strategy can be divided into three main tasks. Firstly, the available preliminary information had to be researched and rated. This research identified loss models based on one dimensional flow physics and empirical correlations as the best suited method to predict the aerodynamic performance. The loss models were calibrated using available performance data to obtain a high prediction quality. As no sufficiently exact models for the prediction of the mechanical loading of the impellercould be identified, a metamodel based on finite element computations was chosen for this estimation. The second task was the development of a sampling method which concentrates samples in regions of the parameter space where high quality designs are predicted by the preliminary information while maintaining a good overall coverage. As available methods like rejection sampling or Markov-chain Monte-Carlo methods did not meet the requirements in terms of sample distribution and input correlation, a new multi-fidelity sampling method called “Filtered Sampling“has been developed. The last task was the development of an automated computational workflow. This workflow encompasses geometry parametrization, geometry generation, grid generation and computation of the aerodynamic performance and the structure mechanic loading. Special emphasis was put into the development of a geometry parametrization strategy based on fluid mechanic considerations to prevent the generation of physically inexpedient designs. Finally, the optimization strategy, which utilizes the previously developed tools, was successfully employed to carry out three optimization tasks. The efficiency of the method was proven by the first and second testcase where an existing compressor design was optimized by the presented method. The results were comparable to optimizations which did not take preliminary information into account, while the required computational expense cloud be halved. In the third testcase, the method was applied to generate a new impeller design. In contrast to the previous examples, this optimization featuredlargervariationsoftheimpellerdesigns. Therefore, theapplicability of the method to parameter spaces with significantly varying designs could be proven, too.

Due to an increased need for hydro-electricity, water storage, and flood protection, it is assumed that a series of new dams will be built throughout the world. Comparing existing design methodologies for arch-type dams, model-based shape optimization can effectively reduce construction costs and leverage the properties of construction materials. To apply the means of shape optimization, suitable variables need to be chosen to formulate the objective function, which is the volume of the arch dam here. In order to increase the consistency with practical conditions, a great number of geometrical and behavioral constraints are included in the mathematical model. An optimization method, namely Genetic Algorithm is adopted which allows a global search.
Traditional optimization techniques are realized based on a deterministic approach, which means that the material properties and loading conditions are assumed to be fixed values. As a result, the real-world structures that are optimized by these approaches suffer from uncertainties that one needs to be aware of. Hence, in any optimization process for arch dams, it is nec- essary to find a methodology that is capable of considering the influences of uncertainties and generating a solution which is robust enough against the uncertainties.
The focus of this thesis is the formulation and the numerical method for the optimization of the arch dam under the uncertainties. The two main models, the probabilistic model, and non-probabilistic models are intro- duced and discussed. Classic procedures of probabilistic approaches un- der uncertainties, such as RDO (robust design optimization) and RBDO (reliability-based design optimization), are in general computationally ex- pensive and rely on estimates of the system’s response variance and fail- ure probabilities. Instead, the robust optimization (RO) method which is based on the non-probabilistic model, will not follow a full probabilistic approach but works with pre-defined confidence levels. This leads to a bi-level optimization program where the volume of the dam is optimized under the worst combination of the uncertain parameters. By this, robust and reliable designs are obtained and the result is independent of any as- sumptions on stochastic properties of the random variables in the model.
The optimization of an arch-type dam is realized here by a robust optimiza- tion method under load uncertainty, where hydraulic and thermal loads are considered. The load uncertainty is modeled as an ellipsoidal expression. Comparing with any traditional deterministic optimization (DO) method, which only concerns the minimum objective value and offers a solution candidate close to limit-states, the RO method provides a robust solution against uncertainties.
All the above mentioned methods are applied to the optimization of the arch dam to compare with the optimal design with DO methods. The re- sults are compared and analyzed to discuss the advantages and drawbacks of each method.
In order to reduce the computational cost, a ranking strategy and an ap- proximation model are further involved to do a preliminary screening. By means of these, the robust design can generate an improved arch dam structure which ensures both safety and serviceability during its lifetime.

Since the Industrial Revolution in the 1700s, the high emission of gaseous wastes into the atmosphere from the usage of fossil fuels has caused a general increase in temperatures globally. To combat the environmental imbalance, there is an increase in the demand for renewable energy sources. Dams play a major role in the generation of “green" energy. However, these structures require frequent and strict monitoring to ensure safe and efficient operation. To tackle the challenges faced in the application of convention dam monitoring techniques, this work proposes the inverse analysis of numerical models to identify damaged regions in the dam. Using a dynamic coupled hydro-mechanical Extended Finite Element Method (XFEM) model and a global optimization strategy, damage (crack) in the dam is identified. By employing seismic waves to probe the dam structure, a more detailed information on the distribution of heterogeneous materials and damaged regions are obtained by the application of the Full Waveform Inversion (FWI) method. The FWI is based on a local optimization strategy and thus it is highly dependent on the starting model. A variety of data acquisition setups are investigated, and an optimal setup is proposed. The effect of different starting models and noise in the measured data on the damage identification is considered. Combining the non-dependence of a starting model of the global optimization strategy based dynamic coupled hydro-mechanical XFEM method and the detailed output of the local optimization strategy based FWI method, an enhanced Full Waveform Inversion is proposed for the structural analysis of dams.

Identification of flaws in structures is a critical element in the management of maintenance and quality assurance processes in engineering. Nondestructive testing (NDT) techniques based on a wide range of physical principles have been developed and are used in common practice for structural health monitoring. However, basic NDT techniques are usually limited in their ability to provide the accurate information on locations, dimensions and shapes of flaws. One alternative to extract additional information from the results of NDT is to append it with a computational model that provides detailed analysis of the physical process involved and enables the accurate identification of the flaw parameters. The aim here is to develop the strategies to uniquely identify cracks in two-dimensional 2D) structures under dynamic loadings.
A local NDT technique combined eXtended Finite Element Method (XFEM) with dynamic loading in order to identify the cracks in the structures quickly and accurately is developed in this dissertation. The Newmark-b time integration method with Rayleigh damping is used for the time integration. We apply Nelder-Mead (NM)and Quasi-Newton (QN) methods for identifying the crack tip in plate. The inverse problem is solved iteratively, in which XFEM is used for solving the forward problem in each iteration. For a timeharmonic excitation with a single frequency and a short-duration signal measured along part of the external boundary, the crack is detected through the solution of an inverse time-dependent problem. Compared to the static load, we show that the dynamic loads are more effective for crack detection problems. Moreover, we tested different dynamic loads and find that NM method works more efficient under the harmonic load than the pounding load while the QN method achieves almost the same results for both load types.
A global strategy, Multilevel Coordinate Search (MCS) with XFEM (XFEM-MCS) methodology under the dynamic electric load, to detect multiple cracks in 2D piezoelectric plates is proposed in this dissertation. The Newmark-b method is employed for the time integration and in each iteration the forward problem is solved by XFEM for various cracks. The objective functional is minimized by using a global search algorithm MCS. The test problems show that the XFEM-MCS algorithm under the dynamic electric load can be effectively employed for multiple cracks detection in piezoelectric materials, and it proves to be robust in identifying defects in piezoelectric structures. Fiber-reinforced composites (FRCs) are extensively applied in practical engineering since they have high stiffness and strength. Experiments reveal a so-called interphase zone, i.e. the space between the outside interface of the fiber and the inside interface of the matrix. The interphase strength between the fiber and the matrix strongly affects the mechanical properties as a result of the large ratio of interface/volume. For the purpose of understanding the mechanical properties of FRCs with functionally graded interphase (FGI), a closed-form expression of the interface strength between a fiber and a matrix is obtained in this dissertation using a continuum modeling approach according to the ver derWaals (vdW) forces. Based on the interatomic potential, we develop a new modified nonlinear cohesive law, which is applied to study the interface delamination of FRCs with FGI under different loadings. The analytical solutions show that the delamination behavior strongly depends on the interphase thickness, the fiber radius, the Young’s moduli and Poisson’s ratios of the fiber and the matrix. Thermal conductivity is the property of a material to conduct heat. With the development and deep research of 2D materials, especially graphene and molybdenum disulfide (MoS2), the thermal conductivity of 2D materials attracts wide attentions. The thermal conductivity of graphene nanoribbons (GNRs) is found to appear a tendency of decreasing under tensile strain by classical molecular dynamics (MD) simulations. Hence, the strain effects of graphene can play a key role in the continuous tunability and applicability of its thermal conductivity property at nanoscale, and the dissipation of thermal conductivity is an obstacle for the applications of thermal management. Up to now, the thermal conductivity of graphene under shear deformation has not been investigated yet. From a practical point of view, good thermal managements of GNRs have significantly potential applications of future GNR-based thermal nanodevices, which can greatly improve performances of the nanosized devices due to heat dissipations. Meanwhile, graphene is a thin membrane structure, it is also important to understand the wrinkling behavior under shear deformation. MoS2 exists in the stable semiconducting 1H phase (1H-MoS2) while the metallic 1T phase (1T-MoS2) is unstable at ambient conditions. As it’s well known that much attention has been focused on studying the nonlinear optical properties of the 1H-MoS2. In a very recent research, the 1T-type monolayer crystals of TMDCs, MX2 (MoS2, WS2 ...) was reported having an intrinsic in-plane negative Poisson’s ratio. Luckily, nearly at the same time, unprecedented long-term (>3months) air stability of the 1T-MoS2 can be achieved by using the donor lithium hydride (LiH). Therefore, it’s very important to study the thermal conductivity of 1T-MoS2.
The thermal conductivity of graphene under shear strain is systematically studied in this dissertation by MD simulations. The results show that, in contrast to the dramatic decrease of thermal conductivity of graphene under uniaxial tensile, the thermal conductivity of graphene is not sensitive to the shear strain, and the thermal conductivity decreases only 12-16%. The wrinkle evolves when the shear strain is around 5%-10%, but the thermal conductivity barely changes.
The thermal conductivities of single-layer 1H-MoS2(1H-SLMoS2) and single-layer 1T-MoS2 (1T-SLMoS2) with different sample sizes, temperatures and strain rates have been studied systematically in this dissertation. We find that the thermal conductivities of 1H-SLMoS2 and 1T-SLMoS2 in both the armchair and the zigzag directions increase with the increasing of the sample length, while the increase of the width of the sample has minor effect on the thermal conductions of these two structures. The thermal conductivity of 1HSLMoS2 is smaller than that of 1T-SLMoS2 under size effect. Furthermore, the temperature effect results show that the thermal conductivities of both 1H-SLMoS2 and 1T-SLMoS2 decrease with the increasing of the temperature. The thermal conductivities of 1HSLMoS2 and 1T-SLMoS2 are nearly the same (difference <6%) in both of the chiral orientations under corresponding temperatures, especially in the armchair direction (difference <2.8%). Moreover, we find that the strain effects on the thermal conductivity of 1HSLMoS2 and 1T-SLMoS2 are different. More specifically, the thermal conductivity decreases with the increasing tensile strain rate for
1T-SLMoS2, while fluctuates with the growth of the strain for 1HSLMoS2. Finally, we find that the thermal conductivity of same sized 1H-SLMoS2 is similar with that of the strained 1H-SLMoS2 structure.

Matrix-free voxel-based finite element method for materials with heterogeneous microstructures
(2018)

Modern image detection techniques such as micro computer tomography
(μCT), magnetic resonance imaging (MRI) and scanning electron microscopy (SEM) provide us with high resolution images of the microstructure of materials in a non-invasive and convenient way. They form the basis for the geometrical models of high-resolution analysis, so called image-based analysis.
However especially in 3D, discretizations of these models reach easily the size of 100 Mill. degrees of freedoms and require extensive hardware resources in terms of main memory and computing power to solve the numerical model. Consequently, the focus of this work is to combine and adapt numerical solution methods to reduce the memory demand first and then the computation time and therewith enable an execution of the image-based analysis on modern computer desktops. Hence, the numerical model is a straightforward grid discretization of the voxel-based (pixels with a third dimension) geometry which omits the boundary detection algorithms and allows reduced storage of the finite element data structure and a matrix-free solution algorithm.
This in turn reduce the effort of almost all applied grid-based solution techniques and results in memory efficient and numerically stable algorithms for the microstructural models. Two variants of the matrix-free algorithm are presented. The efficient iterative solution method of conjugate gradients is used with matrix-free applicable preconditioners such as the Jacobi and the especially suited multigrid method. The jagged material boundaries of the voxel-based mesh are smoothed through embedded boundary elements which contain different material information at the integration point and are integrated sub-cell wise though without additional boundary detection. The efficiency of the matrix-free methods can be retained.

This dissertation is devoted to the theoretical development and experimental laboratory verification of a new damage localization method: The state projection estimation error (SP2E). This method is based on the subspace identification of mechanical structures, Krein space based H-infinity estimation and oblique projections. To explain method SP2E, several theories are discussed and laboratory experiments have been conducted and analysed.
A fundamental approach of structural dynamics is outlined first by explaining mechanical systems based on first principles. Following that, a fundamentally different approach, subspace identification, is comprehensively explained. While both theories, first principle and subspace identification based mechanical systems, may be seen as widespread methods, barely known and new techniques follow up. Therefore, the indefinite quadratic estimation theory is explained. Based on a Popov function approach, this leads to the Krein space based H-infinity theory. Subsequently, a new method for damage identification, namely SP2E, is proposed. Here, the introduction of a difference process, the analysis by its average process power and the application of oblique projections is discussed in depth.
Finally, the new method is verified in laboratory experiments. Therefore, the identification of a laboratory structure at Leipzig University of Applied Sciences is elaborated. Then structural alterations are experimentally applied, which were localized by SP2E afterwards. In the end four experimental sensitivity studies are shown and discussed. For each measurement series the structural alteration was increased, which was successfully tracked by SP2E. The experimental results are plausible and in accordance with the developed theories. By repeating these experiments, the applicability of SP2E for damage localization is experimentally proven.

Polymeric nanocomposites (PNCs) are considered for numerous nanotechnology such as: nano-biotechnology, nano-systems, nanoelectronics, and nano-structured materials. Commonly , they are formed by polymer (epoxy) matrix reinforced with a nanosized filler. The addition of rigid nanofillers to the epoxy matrix has offered great improvements in the fracture toughness without sacrificing other important thermo-mechanical properties. The physics of the fracture in PNCs is rather complicated and is influenced by different parameters. The presence of uncertainty in the predicted output is expected as a result of stochastic variance in the factors affecting the fracture mechanism. Consequently, evaluating the improved fracture toughness in PNCs is a challenging problem.
Artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) have been employed to predict the fracture energy of polymer/particle nanocomposites. The ANN and ANFIS models were constructed, trained, and tested based on a collection of 115 experimental datasets gathered from the literature. The performance evaluation indices of the developed ANN and ANFIS showed relatively small error, with high coefficients of determination (R2), and low root mean square error and mean absolute percentage error.
In the framework for uncertainty quantification of PNCs, a sensitivity analysis (SA) has been conducted to examine the influence of uncertain input parameters on the fracture toughness of polymer/clay nanocomposites (PNCs). The phase-field approach is employed to predict the macroscopic properties of the composite considering six uncertain input parameters. The efficiency, robustness, and repeatability are compared and evaluated comprehensively for five different SA methods.
The Bayesian method is applied to develop a methodology in order to evaluate the performance of different analytical models used in predicting the fracture toughness of polymeric particles nanocomposites. The developed method have considered the model and parameters uncertainties based on different reference data (experimental measurements) gained from the literature. Three analytical models differing in theory and assumptions were examined. The coefficients of variation of the model predictions to the measurements are calculated using the approximated optimal parameter sets. Then, the model selection probability is obtained with respect to the different reference data.
Stochastic finite element modeling is implemented to predict the fracture toughness of polymer/particle nanocomposites. For this purpose, 2D finite element model containing an epoxy matrix and rigid nanoparticles surrounded by an interphase zone is generated. The crack propagation is simulated by the cohesive segments method and phantom nodes. Considering the uncertainties in the input parameters, a polynomial chaos expansion (PCE) surrogate model is construed followed by a sensitivity analysis.

Advances in nanotechnology lead to the development of nano-electro-mechanical systems (NEMS) such as nanomechanical resonators with ultra-high resonant frequencies. The ultra-high-frequency resonators have recently received significant attention for wide-ranging applications such as molecular separation, molecular transportation, ultra-high sensitive sensing, high-frequency signal processing, and biological imaging. It is well known that for micrometer length scale, first-principles technique, the most accurate approach, poses serious limitations for comparisons with experimental studies. For such larger size, classical molecular dynamics (MD) simulations are desirable, which require interatomic potentials. Additionally, a mesoscale method such as the coarse-grained (CG) method is another useful method to support simulations for even larger system sizes.
Furthermore, quasi-two-dimensional (Q2D) materials have attracted intensive research interest due to their many novel properties over the past decades. However, the energy dissipation mechanisms of nanomechanical resonators based on several Q2D materials are still unknown. In this work, the addressed main issues include the development of the CG models for molybdenum disulphide (MoS2), investigation of the mechanism effects on black phosphorus (BP) nanoresonators and the application of graphene nanoresonators. The primary coverage and results of the dissertation are as follows:
Method development. Firstly, a two-dimensional (2D) CG model for single layer MoS2 (SLMoS2) is analytically developed. The Stillinger-Weber (SW) potential for this 2D CG model is further parametrized, in which all SW geometrical parameters are determined analytically according to the equilibrium condition for each individual potential term, while the SW energy parameters are derived analytically based on the valence force field model. Next, the 2D CG model is further simplified to one-dimensional (1D) CG model, which describes the 2D SLMoS2 structure using a 1D chain model. This 1D CG model is applied to investigate the relaxed configuration and the resonant oscillation of the folded SLMoS2. Owning to the simplicity nature of the 1D CG model, the relaxed configuration of the folded SLMoS2 is determined analytically, and the resonant oscillation frequency is derived analytically. Considering the increasing interest in studying the properties of other 2D layered materials, and in particular those in the semiconducting transition metal dichalcogenide class like MoS2, the CG models proposed in current work provide valuable simulation approaches.
Mechanism understanding. Two energy dissipation mechanisms of BP nanoresonators are focused exclusively, i.e. mechanical strain effects and defect effects (including vacancy and oxidation). Vacancy defect is intrinsic damping factor for the quality (Q)-factor, while mechanical strain and oxidation are extrinsic damping factors. Intrinsic dissipation (induced by thermal vibrations) in BP resonators (BPRs) is firstly investigated. Specifically, classical MD simulations are performed to examine the temperature dependence for the Q-factor of the single layer BPR (SLBPR) along the armchair and zigzag directions, where two-step fitting procedure is used to extract the frequency and Q-factor from the kinetic energy time history. The Q-factors of BPRs are evaluated through comparison with those of graphene and MoS2 nanoresonators. Next, effects of mechanical strain, vacancy and oxidation on BP nanoresonators are investigated in turn. Considering the increasing interest in studying the properties of BP, and in particular the lack of theoretical study for the BPRs, the results in current work provide a useful reference.
Application. A novel application for graphene nanoresonators, using them to self-assemble small nanostructures such as water chains, is proposed. All of the underlying physics enabling this phenomenon is elucidated. In particular, by drawing inspiration from macroscale self-assembly using the higher order resonant modes of Chladni plates, classical MD simulations are used to investigate the self-assembly of water molecules using
graphene nanoresonators. An analytic formula for the critical resonant frequency based on the interaction between water molecules and graphene is provided. Furthermore, the properties of the water chains assembled by the graphene nanoresonators are studied.

The phenomenon of aerodynamic instability caused by the wind is usually a major design criterion for long-span cable-supported bridges. If the wind speed exceeds the critical flutter speed of the bridge, this constitutes an Ultimate Limit State. The prediction of the flutter boundary, therefore, requires accurate and robust models. The complexity and uncertainty of models for such engineering problems demand strategies for model assessment. This study is an attempt to use the concepts of sensitivity and uncertainty analyses to assess the aeroelastic instability prediction models for long-span bridges. The state-of-the-art theory concerning the determination of the flutter stability limit is presented. Since flutter is a coupling of aerodynamic forcing with a structural dynamics problem, different types and classes of structural and aerodynamic models can be combined to study the interaction. Here, both numerical approaches and analytical models are utilised and coupled in different ways to assess the prediction quality of the coupled model.

The main categories of wind effects on long span bridge decks are buffeting, flutter, vortex-induced vibrations (VIV) which are often critical for the safety and serviceability of the structure. With the rapid increase of bridge spans, research on controlling wind-induced vibrations of long span bridges has been a problem of great concern.The developments of vibration control theories have led to the wide use of tuned mass dampers (TMDs) which has been proven to be effective for suppressing these vibrations both analytically and experimentally. Fire incidents are also of special interest in the stability and safety of long span bridges due to significant role of the complex phenomenon through triple interaction between the deck with the incoming wind flow and the thermal boundary of the surrounding air.
This work begins with analyzing the buffeting response and flutter instability of three dimensional computational structural dynamics (CSD) models of a cable stayed bridge due to strong wind excitations using ABAQUS finite element commercial software. Optimization and global sensitivity analysis are utilized to target the vertical and torsional vibrations of the segmental deck through considering three aerodynamic parameters (wind attack angle, deck streamlined length and viscous damping of the stay cables). The numerical simulations results in conjunction with the frequency analysis results emphasized the existence of these vibrations and further theoretical studies are possible with a high level of accuracy. Model validation is performed by comparing the results of lift and moment coefficients between the created CSD models and two benchmarks from the literature (flat plate theory) and flat plate by (Xavier and co-authors) which resulted in very good agreements between them. Optimum values of the parameters have been identified. Global sensitivity analysis based on Monte Carlo sampling method was utilized to formulate the surrogate models and calculate the sensitivity indices. The rational effect and the role of each parameter on the aerodynamic stability of the structure were calculated and efficient insight has been constructed for the stability of the long span bridge.
2D computational fluid dynamics (CFD) models of the decks are created with the support of MATLAB codes to simulate and analyze the vortex shedding and VIV of the deck. Three aerodynamic parameters (wind speed, deck streamlined length and dynamic viscosity of the air) are dedicated to study their effects on the kinetic energy of the system and the vortices shapes and patterns. Two benchmarks from the literature (Von Karman) and (Dyrbye and Hansen) are used to validate the numerical simulations of the vortex shedding for the CFD models. A good consent between the results was detected. Latin hypercube experimental
method is dedicated to generate the surrogate models for the kinetic energy of the system and the generated lift forces. Variance based sensitivity analysis is utilized to calculate the main sensitivity indices and the interaction orders for each parameter. The kinetic energy approach performed very well in revealing the rational effect and the role of each parameter in the generation of vortex shedding and predicting the early VIV and the critical wind speed.
Both one-way fluid-structure interaction (one-way FSI) simulations and two-way fluid-structure interaction (two-way FSI) co-simulations for the 2D models of the deck are executed to calculate the shedding frequencies for the associated wind speeds in the lock-in region in addition to the lift and drag coefficients. Validation is executed with the results of (Simiu and Scanlan) and the results of flat plate theory compiled by (Munson and co-authors) respectively. High levels of agreements between all the results were detected. A decrease in the critical wind speed and the shedding frequencies considering (two-way FSI) was identified compared to those obtained in the (one-way FSI). The results from the (two-way FSI) approach predicted appreciable decrease in the lift and drag forces as well as prediction of earlier VIV for lower critical wind speeds and lock-in regions which exist at lower natural frequencies of the system. These conclusions help the designers to efficiently plan and consider for the design and safety of the long span bridge before and after construction.
Multiple tuned mass dampers (MTMDs) system has been applied in the three dimensional CSD models of the cable stayed bridge to analyze their control efficiency in suppressing both wind -induced vertical and torsional vibrations of the deck by optimizing three design parameters (mass ratio, frequency ratio and damping ratio) for the (TMDs) supporting on actual field data and minimax optimization technique in addition to MATLAB codes and Fast Fourier Transform technique. The optimum values of each parameter were identified and validated with two benchmarks from the literature, first with (Wang and co-authors) and then with (Lin and co-authors). The validation procedure detected a good agreement between the results. Box-Behnken experimental method is dedicated to formulate the surrogate models to represent the control efficiency of the vertical and torsional vibrations. Sobol's sensitivity indices are calculated for the design parameters in addition to their interaction orders. The optimization results revealed better performance of the MTMDs in controlling both the vertical and the torsional vibrations for higher mode shapes. Furthermore, the calculated rational effect of each design parameter facilitates to increase the control efficiency of the MTMDs in conjunction with the support of the surrogate models which simplifies the process of analysis for vibration control to a great extent.
A novel structural modification approach has been adopted to eliminate the early coupling between the bending and torsional mode shapes of the cable stayed bridge. Two lateral steel
beams are added to the middle span of the structure. Frequency analysis is dedicated to obtain the natural frequencies of the first eight mode shapes of vibrations before and after the structural modification. Numerical simulations of wind excitations are conducted for the 3D model of the cable stayed bridge. Both vertical and torsional displacements are calculated at the mid span of the deck to analyze the bending and the torsional stiffness of the system before and after the structural modification. The results of the frequency analysis after applying lateral steel beams declared that the coupling between the vertical and torsional mode shapes of vibrations has been removed to larger natural frequencies magnitudes and higher rare critical wind speeds with a high factor of safety.
Finally, thermal fluid-structure interaction (TFSI) and coupled thermal-stress analysis are utilized to identify the effects of transient and steady state heat-transfer on the VIV and fatigue of the deck due to fire incidents. Numerical simulations of TFSI models of the deck are dedicated to calculate the lift and drag forces in addition to determining the lock-in regions once using FSI models and another using TFSI models. Vorticity and thermal fields of three fire scenarios are simulated and analyzed. The benchmark of (Simiu and Scanlan) is used to validate the TFSI models, where a good agreement was manifested between the two results. Extended finite element method (XFEM) is adopted to create 3D models of the cable stayed bridge to simulate the fatigue of the deck considering three fire scenarios. The benchmark of (Choi and Shin) is used to validate the damaged models of the deck in which a good coincide was seen between them. The results revealed that the TFSI models and the coupled thermal-stress models are significant in detecting earlier vortex induced vibration and lock-in regions in addition to predicting damages and fatigue of the deck and identifying the role of wind-induced vibrations in speeding up the damage generation and the collapse of the structure in critical situations.

Phase Field Modeling for Fracture with Applications to Homogeneous and Heterogeneous Materials
(2017)

The thesis presents an implementation including different applications of a variational-based approach for gradient type standard dissipative solids. Phase field model for brittle fracture is an application of the variational-based framework for gradient type solids. This model allows the prediction of different crack topologies and states. Of significant concern is the application of theoretical and numerical formulation of the phase field modeling into the commercial finite element software Abaqus in 2D and 3D. The fully coupled incremental variational formulation of phase field method is implemented by using the UEL and UMAT subroutines of Abaqus. The phase field method
considerably reduces the implementation complexity of fracture problems as it removes the need for numerical tracking of discontinuities in the displacement field that are characteristic of discrete crack methods. This is accomplished by replacing the sharp discontinuities with a scalar damage phase field representing the diffuse crack topology wherein the amount of diffusion is controlled by a regularization parameter. The nonlinear coupled system consisting of the linear momentum equation and a diffusion type equation governing the phase field evolution is solved simultaneously via a Newton-
Raphson approach. Post-processing of simulation results to be used as visualization
module is performed via an additional UMAT subroutine implemented in the standard Abaqus viewer.
In the same context, we propose a simple yet effective algorithm to initiate and propagate cracks in 2D geometries which is independent of both particular constitutive laws and specific element technology and dimension. It consists of a localization limiter in the form of the screened Poisson equation with, optionally, local mesh refinement. A staggered scheme for standard equilibrium and screened Cauchy equations is used. The remeshing part of the algorithm consists of a sequence of mesh subdivision and element erosion steps. Element subdivision is based on edge split operations using a
given constitutive quantity (either damage or void fraction). Mesh smoothing makes use of edge contraction as function of a given constitutive quantity such as the principal stress or void fraction. To assess the robustness and accuracy of this algorithm, we use both quasi-brittle benchmarks and ductile tests.
Furthermore, we introduce a computational approach regarding mechanical loading in microscale on an inelastically deforming composite material. The nanocomposites material of fully exfoliated clay/epoxy is shaped to predict macroscopic elastic and fracture related material parameters based on their fine–scale features. Two different configurations of polymer nanocomposites material (PNCs) have been studied. These configurations are fully bonded PNCs and PNCs with an interphase zone formation between the matrix and the clay reinforcement. The representative volume element of PNCs specimens with different clay weight contents, different aspect ratios, and different
interphase zone thicknesses are generated by adopting Python scripting. Different constitutive models are employed for the matrix, the clay platelets, and the interphase zones. The brittle fracture behavior of the epoxy matrix and the interphase zones material are modeled using the phase field approach, whereas the stiff silicate clay platelets of the composite are designated as a linear elastic material. The comprehensive study investigates the elastic and fracture behavior of PNCs composites, in addition to predict Young’s modulus, tensile strength, fracture toughness, surface energy dissipation, and cracks surface area in the composite for different material parameters, geometry, and interphase zones properties and thicknesses.

The thesis investigates at the computer aided simulation process for operational vibration analysis of complex coupled systems. As part of the internal methods project “Absolute Values” of the BMW Group, the thesis deals with the analysis of the structural dynamic interactions and excitation interactions. The overarching aim of the methods project is to predict the operational vibrations of engines.
Simulations are usually used to analyze technical aspects (e. g. operational vibrations, strength, ...) of single components in the industrial development. The boundary conditions of submodels are mostly based on experiences. So the interactions with neighboring components and systems are neglected. To get physically more realistic results but still efficient simulations, this work wants to support the engineer during the preprocessing phase by useful criteria.
At first suitable abstraction levels based on the existing literature are defined to identify structural dynamic interactions and excitation interactions of coupled systems. So it is possible to separate different effects of the coupled subsystems. On this basis, criteria are derived to assess the influence of interactions between the considered systems. These criteria can be used during the preprocessing phase and help the engineer to build up efficient models with respect to the interactions with neighboring systems. The method was developed by using several models with different complexity levels. Furthermore, the method is proved for the application in the industrial environment by using the example of a current combustion engine.

The gradual digitization in the architecture, engineering, and construction industry over the past fifty years led to an extremely heterogeneous software environment, which today is embodied by the multitude of different digital tools and proprietary data formats used by the many specialists contributing to the design process in a construction project. Though these projects become increasingly complex, the demands on financial efficiency and the completion within a tight schedule grow at the same time. The digital collaboration of project partners has been identified as one key issue in successfully dealing with these challenges. Yet currently, the numerous software applications and their respective individual views on the design process severely impede that collaboration.
An approach to establish a unified basis for the digital collaboration, regardless of the existing software heterogeneity, is a comprehensive digital building model contributed to by all projects partners. This type of data management known as building information modeling (BIM) has many benefits, yet its adoption is associated with many difficulties and thus, proceeds only slowly. One aspect in the field of conflicting requirements on such a digital model is the cooperation of architects and structural engineers. Traditionally, these two disciplines use different abstractions of reality for their models that in consequence lead to incompatible digital representations thereof.
The onset of isogeometric analysis (IGA) promised to ease the discrepancy in design and analysis model representations. Yet, that initial focus quickly shifted towards using these methods as a more powerful basis for numerical simulations. Furthermore, the isogeometric representation alone is not capable of solving the model abstraction problem. It is thus the intention of this work to contribute to an improved digital collaboration of architects and engineers by exploring an integrated analysis approach on the basis of an unified digital model and solid geometry expressed by splines. In the course of this work, an analysis framework is developed that utilizes such models to automatically conduct numerical simulations commonly required in construction projects. In essence, this allows to retrieve structural analysis results from BIM models in a fast and simple manner, thereby facilitating rapid design iterations and profound design feedback.
The BIM implementation Industry Foundation Classes (IFC) is reviewed with regard to its capabilities of representing the unified model. The current IFC schema strongly supports the use of redundant model data, a major pitfall in digital collaboration. Additionally, it does not allow to describe the geometry by volumetric splines. As the pursued approach builds upon a unique model for both, architectural and structural design, and furthermore requires solid geometry, necessary schema modifications are suggested.
Structural entities are modeled by volumetric NURBS patches, each of which constitutes an individual subdomain that, with regard to the analysis, is incompatible with the remaining full model. The resulting consequences for numerical simulation are elaborated in this work. The individual subdomains have to be weakly coupled, for which the mortar method is used. Different approaches to discretize the interface traction fields are implemented and their respective impact on the analysis results is evaluated. All necessary coupling conditions are automatically derived from the related geometry model.
The weak coupling procedure leads to a linear system of equations in saddle point form, which, owed to the volumetric modeling, is large in size and, the associated coefficient matrix has, due to the use of higher degree basis functions, a high bandwidth. The peculiarities of the system require adapted solution methods that generally cause higher numerical costs than the standard procedures for symmetric, positive-definite systems do. Different methods to solve the specific system are investigated and an efficient parallel algorithm is finally proposed.
When the structural analysis model is derived from the unified model in the BIM data, it does in general initially not meet the requirements on the discretization that are necessary to obtain sufficiently accurate analysis results. The consequently necessary patch refinements must be controlled automatically to allowfor an entirely automatic analysis procedure. For that purpose, an empirical refinement scheme based on the geometrical and possibly mechanical properties of the specific entities is proposed. The level of refinement may be selectively manipulated by the structural engineer in charge. Furthermore, a Zienkiewicz-Zhu type error estimator is adapted for the use with isogeometric analysis results. It is shown that also this estimator can be used to steer an adaptive refinement procedure.

Piezoelectric materials are used in several applications as sensors and actuators where they experience high stress and electric field concentrations as a result of which they may fail due to fracture. Though there are many analytical and experimental works on piezoelectric fracture mechanics. There are very few studies about damage detection, which is an interesting way to prevent the failure of these ceramics.
An iterative method to treat the inverse problem of detecting cracks and voids in piezoelectric structures is proposed. Extended finite element method (XFEM) is employed for solving the inverse problem as it allows the use of a single regular mesh for large number of iterations with different flaw geometries.
Firstly, minimization of cost function is performed by Multilevel Coordinate Search (MCS) method. The XFEM-MCS methodology is applied to two dimensional electromechanical problems where flaws considered are straight cracks and elliptical voids. Then a numerical method based on combination of classical shape derivative and level set method for front propagation used in structural optimization is utilized to minimize the cost function. The results obtained show that the XFEM-level set methodology is effectively able to determine the number of voids in a piezoelectric structure and its corresponding locations.
The XFEM-level set methodology is improved to solve the inverse problem of detecting inclusion interfaces in a piezoelectric structure. The material interfaces are implicitly represented by level sets which are identified by applying regularisation using total variation penalty terms. The formulation is presented for three dimensional structures and inclusions made of different materials are detected by using multiple level sets. The results obtained prove that the iterative procedure proposed can determine the location and approximate shape of material subdomains in the presence of higher noise levels.
Piezoelectric nanostructures exhibit size dependent properties because of surface elasticity and surface piezoelectricity. Initially a study to understand the influence of surface elasticity on optimization of nano elastic beams is performed. The boundary of the nano structure is implicitly represented by a level set function, which is considered as the design variable in the optimization process. Two objective functions, minimizing the total potential energy of a nanostructure subjected to a material volume constraint and minimizing the least square error compared to a target
displacement, are chosen for the numerical examples. The numerical examples demonstrate the importance of size and aspect ratio in determining how surface effects impact the optimized topology of nanobeams.
Finally a conventional cantilever energy harvester with a piezoelectric nano layer is analysed. The presence of surface piezoelectricity in nano beams and nano plates leads to increase in electromechanical coupling coefficient. Topology optimization of these piezoelectric structures in an energy harvesting device to further increase energy conversion using appropriately modified XFEM-level set algorithm is performed .

Briefly, the two basic questions that this research is supposed to answer are:
1. Howmuch fiber is needed and how fibers should be distributed through a fiber reinforced composite (FRC) structure in order to obtain the optimal and reliable structural response?
2. How do uncertainties influence the optimization results and reliability of the structure?
Giving answer to the above questions a double stage sequential optimization algorithm for finding the optimal content of short fiber reinforcements and their distribution in the composite structure, considering uncertain design parameters, is presented. In the first stage, the optimal amount of short fibers in a FRC structure with uniformly distributed fibers is conducted in the framework of a Reliability Based Design Optimization (RBDO) problem. Presented model considers material, structural and modeling uncertainties. In the second stage, the fiber distribution optimization (with the aim to further increase in structural reliability) is performed by defining a fiber distribution function through a Non-Uniform Rational BSpline (NURBS) surface. The advantages of using the NURBS surface as a fiber distribution function include: using the same data set for the optimization and analysis; high convergence rate due to the smoothness of the NURBS; mesh independency of the optimal layout; no need for any post processing technique and its non-heuristic nature. The output of stage 1 (the optimal fiber content for homogeneously distributed fibers) is considered as the input of stage 2. The output of stage 2 is the Reliability Index (b ) of the structure with the optimal fiber content and distribution.
First order reliability method (in order to approximate the limit state function) as well as different material models including Rule of Mixtures, Mori-Tanaka, energy-based approach and stochastic multi-scales are implemented in different examples. The proposed combined model is able to capture the role of available uncertainties in FRC structures through a computationally efficient algorithm using all sequential, NURBS and sensitivity based techniques. The methodology is successfully implemented for interfacial shear stress optimization in sandwich beams and also for optimization of the internal cooling channels in a ceramic matrix composite.
Finally, after some changes and modifications by combining Isogeometric Analysis, level set and point wise density mapping techniques, the computational framework is extended for topology optimization of piezoelectric / flexoelectric materials.

The key objective of this research is to study fracture with a meshfree method, local maximum entropy approximations, and model fracture in thin shell structures with complex geometry and topology. This topic is of high relevance for real-world applications, for example in the automotive industry and in aerospace engineering. The shell structure can be described efficiently by meshless methods which are capable of describing complex shapes as a collection of points instead of a structured mesh. In order to find the appropriate numerical method to achieve this goal, the first part of the work was development of a method based on local maximum entropy (LME)
shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method (XFEM) at a comparable computational cost. In addition, we keep the advantages of the LME shape functions,such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes
near the crack such as blending or shifting.
As extension of this method to three dimensional problems and complex thin shell structures with arbitrary crack growth is cumbersome, we developed a phase field model for fracture using LME. Phase field models provide a powerful tool to tackle moving interface problems, and have been extensively used in physics and materials science. Phase methods are gaining popularity in a wide set of applications in applied science and engineering, recently a second order phase field approximation for brittle fracture has gathered significant interest in computational fracture such that sharp cracks discontinuities are modeled by a diffusive crack. By minimizing the system energy with respect to the mechanical displacements and the phase-field, subject to an irreversibility condition to avoid crack healing, this model can describe crack nucleation, propagation, branching and merging. One of the main advantages of the phase field modeling of fractures is the unified treatment of the interfacial tracking and mechanics, which potentially leads to simple, robust, scalable computer codes applicable to complex systems. In other words, this approximation reduces considerably the implementation complexity because the numerical tracking of the fracture is not needed, at the expense of a high computational cost. We present a fourth-order phase field model for fracture based on local maximum entropy (LME) approximations. The higher order continuity of the meshfree LME approximation allows to directly solve the fourth-order phase field equations without splitting the fourth-order differential equation into two second order differential equations. Notably, in contrast to previous discretizations that use at least a quadratic basis, only linear completeness is needed in the LME approximation. We show that the crack surface can be captured more accurately in the fourth-order model than the second-order model. Furthermore, less nodes are needed for the fourth-order model to resolve the crack path. Finally, we demonstrate the performance of the proposed meshfree fourth order phase-field formulation for 5 representative numerical examples. Computational results will be compared to analytical solutions within linear elastic fracture mechanics and experimental data for three-dimensional crack propagation.
In the last part of this research, we present a phase-field model for fracture in Kirchoff-Love thin shells using the local maximum-entropy (LME) meshfree method. Since the crack is a natural outcome of the analysis it does not require an explicit representation and tracking, which is advantageous over techniques as the extended finite element method that requires tracking of the crack paths. The geometric description of the shell is based on statistical learning techniques that allow dealing with general point set surfaces avoiding a global parametrization, which can be applied to tackle surfaces of complex geometry and topology. We show the flexibility and robustness of the present methodology for two examples: plate in tension and a set of open connected
pipes.

Nanostructured materials are extensively applied in many fields of material science for new industrial applications, particularly in the automotive, aerospace industry due to their exceptional physical and mechanical properties. Experimental testing of nanomaterials is expensive, timeconsuming,challenging and sometimes unfeasible. Therefore,computational simulations have been employed as alternative method to predict macroscopic material properties. The behavior of polymeric nanocomposites (PNCs) are highly complex.
The origins of macroscopic material properties reside in the properties and interactions taking place on finer scales. It is therefore essential to use multiscale modeling strategy to properly account for all large length and time scales associated with these material systems, which across many orders of magnitude. Numerous multiscale models of PNCs have been established, however, most of them connect only two scales. There are a few multiscale models for PNCs bridging four length scales (nano-, micro-, meso- and macro-scales). In addition, nanomaterials are stochastic in nature and the prediction of macroscopic mechanical properties are influenced by many factors such as fine-scale features. The predicted mechanical properties obtained by traditional approaches significantly deviate from the measured values in experiments due to neglecting uncertainty of material features. This discrepancy is indicated that the effective macroscopic properties of materials are highly sensitive to various sources of uncertainty, such as loading and boundary conditions and material characteristics, etc., while very few stochastic multiscale models for PNCs have been developed. Therefore, it is essential to construct PNC models within the framework of stochastic modeling and quantify the stochastic effect of the input parameters on the macroscopic mechanical properties of those materials.
This study aims to develop computational models at four length scales (nano-, micro-, meso- and macro-scales) and hierarchical upscaling approaches bridging length scales from nano- to macro-scales. A framework for uncertainty quantification (UQ) applied to predict the mechanical properties
of the PNCs in dependence of material features at different scales is studied. Sensitivity and uncertainty analysis are of great helps in quantifying the effect of input parameters, considering both main and interaction effects, on the mechanical properties of the PNCs. To achieve this major
goal, the following tasks are carried out:
At nano-scale, molecular dynamics (MD) were used to investigate deformation mechanism of glassy amorphous polyethylene (PE) in dependence of temperature and strain rate. Steered molecular dynamics (SMD)were also employed to investigate interfacial characteristic of the PNCs.
At mico-scale, we developed an atomistic-based continuum model represented by a representative volume element (RVE) in which the SWNT’s properties and the SWNT/polymer interphase are modeled at nano-scale, the surrounding polymer matrix is modeled by solid elements. Then, a two-parameter model was employed at meso-scale. A hierarchical multiscale approach has been developed to obtain the structure-property relations at one length scale and transfer the effect to the higher length
scales. In particular, we homogenized the RVE into an equivalent fiber.
The equivalent fiber was then employed in a micromechanical analysis (i.e. Mori-Tanaka model) to predict the effective macroscopic properties of the PNC. Furthermore, an averaging homogenization process was also used to obtain the effective stiffness of the PCN at meso-scale.
Stochastic modeling and uncertainty quantification consist of the following ingredients:
- Simple random sampling, Latin hypercube sampling, Sobol’ quasirandom sequences, Iman and Conover’s method (inducing correlation in Latin hypercube sampling) are employed to generate independent and dependent sample data, respectively.
- Surrogate models, such as polynomial regression, moving least squares (MLS), hybrid method combining polynomial regression and MLS, Kriging regression, and penalized spline regression, are employed as an approximation of a mechanical model. The advantage of the surrogate models is the high computational efficiency and robust as they can be constructed from a limited amount of available data.
- Global sensitivity analysis (SA) methods, such as variance-based methods for models with independent and dependent input parameters, Fourier-based techniques for performing variance-based methods and partial derivatives, elementary effects in the context of local SA, are used to quantify the effects of input parameters and their interactions on the mechanical properties of the PNCs. A bootstrap technique is used to assess the robustness of the global SA methods with respect to their performance.
In addition, the probability distribution of mechanical properties are determined by using the probability plot method. The upper and lower bounds of the predicted Young’s modulus according to 95 % prediction intervals were provided.
The above-mentioned methods study on the behaviour of intact materials. Novel numerical methods such as a node-based smoothed extended finite element method (NS-XFEM) and an edge-based smoothed phantom node method (ES-Phantom node) were developed for fracture problems. These methods can be used to account for crack at macro-scale for future works. The predicted mechanical properties were validated and verified. They show good agreement with previous experimental and simulations results.

Methods based on B-splines for model representation, numerical analysis and image registration
(2015)

The thesis consists of inter-connected parts for modeling and analysis using newly developed isogeometric methods. The main parts are reproducing kernel triangular B-splines, extended isogeometric analysis for solving weakly discontinuous problems, collocation methods using superconvergent points, and B-spline basis in image registration applications.
Each topic is oriented towards application of isogeometric analysis basis functions to ease the process of integrating the modeling and analysis phases of simulation.
First, we develop reproducing a kernel triangular B-spline-based FEM for solving PDEs. We review the triangular B-splines and their properties. By definition, the triangular basis function is very flexible in modeling complicated domains. However, instability results when it is applied for analysis. We modify the triangular B-spline by a reproducing kernel technique, calculating a correction term for the triangular kernel function from the chosen surrounding basis. The improved triangular basis is capable to obtain the results with higher accuracy and almost optimal convergence rates.
Second, we propose an extended isogeometric analysis for dealing with weakly discontinuous problems such as material interfaces. The original IGA is combined with XFEM-like enrichments which are continuous functions themselves but with discontinuous derivatives. Consequently, the resulting solution space can approximate solutions with weak discontinuities. The method is also applied to curved material interfaces, where the inverse mapping and the curved triangular elements are considered.
Third, we develop an IGA collocation method using superconvergent points. The collocation methods are efficient because no numerical integration is needed. In particular when higher polynomial basis applied, the method has a lower computational cost than Galerkin methods. However, the positions of the collocation points are crucial for the accuracy of the method, as they affect the convergent rate significantly. The proposed IGA collocation method uses superconvergent points instead of the traditional Greville abscissae points. The numerical results show the proposed method can have better accuracy and optimal convergence rates, while the traditional IGA collocation has optimal convergence only for even polynomial degrees.
Lastly, we propose a novel dynamic multilevel technique for handling image registration. It is application of the B-spline functions in image processing. The procedure considered aims to align a target image from a reference image by a spatial transformation. The method starts with an energy function which is the same as a FEM-based image registration. However, we simplify the solving procedure, working on the energy function directly. We dynamically solve for control points which are coefficients of B-spline basis functions. The new approach is more simple and fast. Moreover, it is also enhanced by a multilevel technique in order to prevent instabilities. The numerical testing consists of two artificial images, four real bio-medical MRI brain and CT heart images, and they show our registration method is accurate, fast and efficient, especially for large deformation problems.

One major research focus in the Material Science and Engineering Community in the past decade has been to obtain a more fundamental understanding on the phenomenon 'material failure'. Such an understanding is critical for engineers and scientists developing new materials with higher strength and toughness, developing robust designs against failure, or for those concerned with an accurate estimate of a component's design life. Defects like cracks and dislocations evolve at
nano scales and influence the macroscopic properties such as strength, toughness and ductility of a material. In engineering applications, the global response of the system is often governed by the behaviour at the smaller length scales. Hence, the sub-scale behaviour must be computed accurately for good predictions of the full scale behaviour.
Molecular Dynamics (MD) simulations promise to reveal the fundamental mechanics of material failure by modeling the atom to atom interactions. Since the atomistic dimensions are of the order of Angstroms ( A), approximately 85 billion atoms are required to model a 1 micro- m^3 volume of Copper. Therefore, pure atomistic models are prohibitively expensive with everyday engineering computations involving macroscopic cracks and shear bands, which are much larger than the atomistic length and time scales. To reduce the computational effort, multiscale methods are required, which are able to couple a continuum description of the structure with an atomistic description. In such paradigms, cracks and dislocations are explicitly modeled at the atomistic scale, whilst a self-consistent continuum model elsewhere.
Many multiscale methods for fracture are developed for "fictitious" materials based on "simple" potentials such as the Lennard-Jones potential. Moreover, multiscale methods for evolving cracks are rare. Efficient methods to coarse grain the fine scale defects are missing. However, the existing multiscale methods for fracture do not adaptively adjust the fine scale domain as the crack propagates. Most methods, therefore only "enlarge" the fine scale domain and therefore drastically increase computational cost. Adaptive adjustment requires the fine scale domain to be refined and coarsened. One of the major difficulties in multiscale methods for fracture is to up-scale fracture related material information from the fine scale to the coarse scale, in particular for complex crack problems. Most of the existing approaches therefore were applied to examples with comparatively few macroscopic cracks.
Key contributions
The bridging scale method is enhanced using the phantom node method so that cracks can be modeled at the coarse scale. To ensure self-consistency in the bulk, a virtual atom cluster is devised providing the response of the intact material at the coarse scale. A molecular statics model is employed in the fine scale where crack propagation is modeled by naturally breaking the bonds. The fine scale and coarse scale models are coupled by enforcing the displacement boundary conditions on the ghost atoms. An energy criterion is used to detect the crack tip location. Adaptive refinement and coarsening schemes are developed and implemented during the crack propagation. The results were observed to be in excellent agreement with the pure atomistic simulations. The developed multiscale method is one of the first adaptive multiscale method for fracture.
A robust and simple three dimensional coarse graining technique to convert a given atomistic region into an equivalent coarse region, in the context of multiscale fracture has been developed. The developed method is the first of its kind. The developed coarse graining technique can be applied to identify and upscale the defects like: cracks, dislocations and shear bands. The current method has been applied to estimate the equivalent coarse scale models of several complex fracture patterns arrived from the pure atomistic simulations. The upscaled fracture pattern agree well with the actual fracture pattern. The error in the potential energy of the pure atomistic and the coarse grained model was observed to be acceptable.
A first novel meshless adaptive multiscale method for fracture has been developed. The phantom node method is replaced by a meshless differential reproducing kernel particle method. The differential reproducing kernel particle method is comparatively more expensive but allows for a more "natural" coupling between the two scales due to the meshless interpolation functions. The higher order continuity is also beneficial. The centro symmetry parameter is used to detect the crack tip location. The developed multiscale method is employed to study the complex crack propagation. Results based on the meshless adaptive multiscale method were observed to be in excellent agreement with the pure atomistic simulations.
The developed multiscale methods are applied to study the fracture in practical materials like Graphene and Graphene on Silicon surface. The bond stretching and the bond reorientation were observed to be the net mechanisms of the crack growth in Graphene. The influence of time step on the crack propagation was studied using two different time steps. Pure atomistic simulations of fracture in Graphene on Silicon surface are presented. Details of the three dimensional multiscale method to study the fracture in Graphene on Silicon surface are discussed.

Structural vibration control of high-speed railway bridges using tuned mass dampers, semi-active tuned mass dampers, fluid viscous dampers and magnetorheological dampers to reduce resonant structural vibrations is studied. In this work, the addressed main issues include modeling of the dynamic interaction of the structures, optimization of the parameters of the dampers and comparison of their efficiency.
A new approach to optimize multiple tuned mass damper systems on an uncertain model is proposed based on the H-infinity optimization criteria and the DK iteration procedure with norm-bounded uncertainties in frequency domain. The parameters of tuned mass dampers are optimized directly and simultaneously on different modes contributing significantly to the multi-resonant peaks to explore the different possible combinations of parameters. The effectiveness of the present method is also evaluated through comparison with a previous method.
In the case of semi-active tuned mass dampers, an optimization algorithm is derived to control the magnetorheological damper in these semi-active damping systems. The use of the proposed algorithm can generate various combinations of control gains and state variables. This can lead to the improvement of the ability of MR dampers to track the desired control forces. An uncertain model to reduce detuning effects is also considered in this work.
Next, for fluid viscous dampers, in order to tune the optimal parameters of fluid viscous dampers to the vicinity of the exact values, analytical formulae which can include structural damping are developed based on the perturbation method. The proposed formulae can also be considered as an improvement of the previous analytical formulae, especially for bridge beams with large structural damping.
Finally, a new combination of magnetorheological dampers and a double-beam system to improve the performance of the primary structure vibration is proposed. An algorithm to control magnetorheological dampers in this system is developed by using standard linear matrix inequality techniques. Weight functions as a loop shaping procedure are also introduced in the feedback controllers to improve the tracking ability of magnetorheological damping forces. To this end, the effectiveness of magnetorheological dampers controlled by the proposed scheme, along with the effects of the uncertain and time-delay parameters on the models, are evaluated through numerical simulations.
Additionally, a comparison of the dampers based on their performance is also considered in this work.

Environmental and operational variables and their impact on structural responses have been acknowledged as one of the most important challenges for the application of the ambient vibration-based damage identification in structures. The damage detection procedures may yield poor results, if the impacts of loading and environmental conditions of the structures are not considered.
The reference-surface-based method, which is proposed in this thesis, is addressed to overcome this problem. In the proposed method, meta-models are used to take into account significant effects of the environmental and operational variables. The usage of the approximation models, allows the proposed method to simply handle multiple non-damaged variable effects simultaneously, which for other methods seems to be very complex. The input of the meta-model are the multiple non-damaged variables while the output is a damage indicator.
The reference-surface-based method diminishes the effect of the non-damaged variables to the vibration based damage detection results. Hence, the structure condition that is assessed by using ambient vibration data at any time would be more reliable. Immediate reliable information regarding the structure condition is required to quickly respond to the event, by means to take necessary actions concerning the future use or further investigation of the structures, for instance shortly after extreme events such as earthquakes.
The critical part of the proposed damage detection method is the learning phase, where the meta-models are trained by using input-output relation of observation data. Significant problems that may encounter during the learning phase are outlined and some remedies to overcome the problems are suggested.
The proposed damage identification method is applied to numerical and experimental models. In addition to the natural frequencies, wavelet energy and stochastic subspace damage indicators are used.

This thesis concerns the physical and mechanical interactions on carbon nanotubes and polymers by multiscale modeling. CNTs have attracted considerable interests in view of their unique mechanical, electronic, thermal, optical and structural properties, which enable them to have many potential applications.
Carbon nanotube exists in several structure forms, from individual single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) to carbon nanotube bundles and networks. The mechanical properties of SWCNTs and MWCNTs have been extensively studied by continuum modeling and molecular dynamics (MD) simulations in the past decade since the properties could be important in the CNT-based devices. CNT bundles and networks feature outstanding mechanical performance and hierarchical structures and network topologies, which have been taken as a potential saving-energy material. In the synthesis of nanocomposites, the formation of the CNT bundles and networks is a challenge to remain in understanding how to measure and predict the properties of such large systems. Therefore, a mesoscale method such as a coarse-grained (CG) method should be developed to study the nanomechanical characterization of CNT bundles and networks formation.
In this thesis, the main contributions can be written as follows: (1) Explicit solutions for the cohesive energy between carbon nanotubes, graphene and substrates are obtained through continuum modeling of the van der Waals interaction between them. (2) The CG potentials of SWCNTs are established by a molecular mechanics model. (3) The binding energy between two parallel and crossing SWCNTs and MWCNTs is obtained by continuum modeling of the van der Waals interaction between them. Crystalline and amorphous polymers are increasingly used in modern industry as tructural materials due to its important mechanical and physical properties. For crystalline polyethylene (PE), despite its importance and the studies of available MD simulations and continuum models, the link between molecular and continuum descriptions of its mechanical properties is still not well established. For amorphous polymers, the chain length and temperature effect on their
elastic and elastic-plastic properties has been reported based on the united-atom (UA) and CG MD imulations in our previous work. However, the effect of the CL and temperature on the failure behavior is not understood well yet. Especially, the failure behavior under shear has been scarcely reported in previous work. Therefore, understanding the molecular origins of macroscopic fracture behavior such as fracture energy is a fundamental scientific challenge.
In this thesis, the main contributions can be written as follows: (1) An analytical molecular mechanics model is developed to obtain the size-dependent elastic properties of crystalline PE.
(2) We show that the two molecular mechanics models, the stick-spiral and the beam models, predict considerably different mechanical properties of materials based on energy equivalence. The difference between the two models is independent of the materials. (3) The tensile and shear failure behavior dependence on chain length and temperature in amorphous polymers are scrutinized using molecular dynamics simulations. Finally, the influence of polymer wrapped two neighbouring SWNTs’ dispersion on their load transfer is investigated by molecular dynamics (MD) simulations, in which the SWNTs' position, the polymer chain length and the temperature on the interaction force is systematically studied.

Modern digital material approaches for the visualization and simulation of heterogeneous materials allow to investigate the behavior of complex multiphase materials with their physical nonlinear material response at various scales. However, these computational techniques require extensive hardware resources with respect to computing power and main memory to solve numerically large-scale discretized models in 3D. Due to a very high number of degrees of freedom, which may rapidly be increased to the two-digit million range, the limited hardware ressources are to be utilized in a most efficient way to enable an execution of the numerical algorithms in minimal computation time. Hence, in the field of computational mechanics, various methods and algorithms can lead to an optimized runtime behavior of nonlinear simulation models, where several approaches are proposed and investigated in this thesis.
Today, the numerical simulation of damage effects in heterogeneous materials is performed by the adaption of multiscale methods. A consistent modeling in the three-dimensional space with an appropriate discretization resolution on each scale (based on a hierarchical or concurrent multiscale model), however, still contains computational challenges in respect to the convergence behavior, the scale transition or the solver performance of the weak coupled problems. The computational efficiency and the distribution among available hardware resources (often based on a parallel hardware architecture) can significantly be improved. In the past years, high-performance computing (HPC) and graphics processing unit (GPU) based computation techniques were established for the investigationof scientific objectives. Their application results in the modification of existing and the development of new computational methods for the numerical implementation, which enables to take advantage of massively clustered computer hardware resources. In the field of numerical simulation in material science, e.g. within the investigation of damage effects in multiphase composites, the suitability of such models is often restricted by the number of degrees of freedom (d.o.f.s) in the three-dimensional spatial discretization. This proves to be difficult for the type of implementation method used for the nonlinear simulation procedure and, simultaneously has a great influence on memory demand and computational time.
In this thesis, a hybrid discretization technique has been developed for the three-dimensional discretization of a three-phase material, which is respecting the numerical efficiency of nonlinear (damage) simulations of these materials. The increase of the computational efficiency is enabled by the improved scalability of the numerical algorithms. Consequently, substructuring methods for partitioning the hybrid mesh were implemented, tested and adapted to the HPC computing framework using several hundred CPU (central processing units) nodes for building the finite element assembly. A memory-efficient iterative and parallelized equation solver combined with a special preconditioning technique for solving the underlying equation system was modified and adapted to enable combined CPU and GPU based computations.
Hence, it is recommended by the author to apply the substructuring method for hybrid meshes, which respects different material phases and their mechanical behavior and which enables to split the structure in elastic and inelastic parts. However, the consideration of the nonlinear material behavior, specified for the corresponding phase, is limited to the inelastic domains only, and by that causes a decreased computing time for the nonlinear procedure. Due to the high numerical effort for such simulations, an alternative approach for the nonlinear finite element analysis, based on the sequential linear analysis, was implemented in respect to scalable HPC. The incremental-iterative procedure in finite element analysis (FEA) during the nonlinear step was then replaced by a sequence of linear FE analysis when damage in critical regions occured, known in literature as saw-tooth approach. As a result, qualitative (smeared) crack initiation in 3D multiphase specimens has efficiently been simulated.

Numerical models and their combination with advanced solution strategies are standard tools for many engineering disciplines to design or redesign structures and to optimize designs with the purpose to improve specific requirements. As the successful application of numerical models depends on their suitability to represent the behavior related to the intended use, they should be validated by experimentally obtained results. If the discrepancy between numerically derived and experimentally obtained results is not acceptable, a model revision or a revision of the experiment need to be considered. Model revision is divided into two classes, the model updating and the basic revision of the numerical model. The presented thesis is related to a special branch of model updating, the vibration-based model updating. Vibration-based model updating is a tool to improve the correlation of the numerical model by adjusting uncertain model input parameters by means of results extracted from vibration tests. Evidently, uncertainties related to the experiment, the numerical model, or the applied numerical solving strategies can influence the correctness of the identified model input parameters. The reduction of uncertainties for two critical problems and the quantification of uncertainties related to the investigation of several nominally identical structures are the main emphases of this thesis. First, the reduction of uncertainties by optimizing reference sensor positions is considered. The presented approach relies on predicted power spectral amplitudes and an initial finite element model as a basis to define the assessment criterion for predefined sensor positions. In combination with geometry-based design variables, which represent the sensor positions, genetic and particle swarm optimization algorithms are applied. The applicability of the proposed approach is demonstrated on a numerical benchmark study of a simply supported beam and a case study of a real test specimen. Furthermore, the theory of determining the predicted power spectral amplitudes is validated with results from vibration tests. Second, the possibility to reduce uncertainties related to an inappropriate assignment for numerically derived and experimentally obtained modes is investigated. In the context of vibration-based model updating, the correct pairing is essential. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches. Hence, an alternative criterion, the energy-based modal assurance criterion, is proposed. This criterion combines the mathematical characteristic of orthogonality with the physical properties of the structure by modal strain energies. A numerical example and a case study with experimental data are presented to show the advantages of the proposed energy-based modal assurance criterion in comparison to the traditional modal assurance criterion. Third, the application of optimization strategies combined with information theory based objective functions is analyzed for the purpose of stochastic model updating. This approach serves as an alternative to the common sensitivity-based stochastic model updating strategies. Their success depends strongly on the defined initial model input parameters. In contrast, approaches based on optimization strategies can be more flexible. It can be demonstrated, that the investigated nature inspired optimization strategies in combination with Bhattacharyya distance and Kullback-Leibler divergence are appropriate. The obtained accuracies and the respective computational effort are comparable with sensitivity-based stochastic model updating strategies. The application of model updating procedures to improve the quality and suitability of a numerical model is always related to additional costs. The presented innovative approaches will contribute to reduce and quantify uncertainties within a vibration-based model updating process. Therefore, the increased benefit can compensate the additional effort, which is necessary to apply model updating procedures.

The numerical simulation of damage using phenomenological models on the macroscale was state of the art for many decades. However, such models are not able to capture the complex nature of damage, which simultaneously proceeds on multiple length scales. Furthermore, these phenomenological models usually contain damage parameters, which are physically not interpretable. Consequently, a reasonable experimental determination of these parameters is often impossible. In the last twenty years, the ongoing advance in computational capacities provided new opportunities for more and more detailed studies of the microstructural damage behavior. Today, multiphase models with several million degrees of freedom enable for the numerical simulation of micro-damage phenomena in naturally heterogeneous materials. Therewith, the application of multiscale concepts for the numerical investigation of the complex nature of damage can be realized. The presented thesis contributes to a hierarchical multiscale strategy for the simulation of brittle intergranular damage in polycrystalline materials, for example aluminum. The numerical investigation of physical damage phenomena on an atomistic microscale and the integration of these physically based information into damage models on the continuum meso- and macroscale is intended. Therefore, numerical methods for the damage analysis on the micro- and mesoscale including the scale transfer are presented and the transition to the macroscale is discussed. The investigation of brittle intergranular damage on the microscale is realized by the application of the nonlocal Quasicontinuum method, which fully describes the material behavior by atomistic potential functions, but reduces the number of atomic degrees of freedom by introducing kinematic couplings. Since this promising method is applied only by a limited group of researchers for special problems, necessary improvements have been realized in an own parallelized implementation of the 3D nonlocal Quasicontinuum method. The aim of this implementation was to develop and combine robust and efficient algorithms for a general use of the Quasicontinuum method, and therewith to allow for the atomistic damage analysis in arbitrary grain boundary configurations. The implementation is applied in analyses of brittle intergranular damage in ideal and nonideal grain boundary models of FCC aluminum, considering arbitrary misorientations. From the microscale simulations traction separation laws are derived, which describe grain boundary decohesion on the mesoscale. Traction separation laws are part of cohesive zone models to simulate the brittle interface decohesion in heterogeneous polycrystal structures. 2D and 3D mesoscale models are presented, which are able to reproduce crack initiation and propagation along cohesive interfaces in polycrystals. An improved Voronoi algorithm is developed in 2D to generate polycrystal material structures based on arbitrary distribution functions of grain size. The new model is more flexible in representing realistic grain size distributions. Further improvements of the 2D model are realized by the implementation and application of an orthotropic material model with Hill plasticity criterion to grains. The 2D and 3D polycrystal models are applied to analyze crack initiation and propagation in statically loaded samples of aluminum on the mesoscale without the necessity of initial damage definition.

The nonlinear behavior of concrete can be attributed to the propagation of microcracks within the heterogeneous internal material structure. In this thesis, a mesoscale model is developed which allows for the explicit simulation of these microcracks. Consequently, the actual physical phenomena causing the complex nonlinear macroscopic behavior of concrete can be represented using rather simple material formulations. On the mesoscale, the numerical model explicitly resolves the components of the internal material structure. For concrete, a three-phase model consisting of aggregates, mortar matrix and interfacial transition zone is proposed. Based on prescribed grading curves, an efficient algorithm for the generation of three-dimensional aggregate distributions using ellipsoids is presented. In the numerical model, tensile failure of the mortar matrix is described using a continuum damage approach. In order to reduce spurious mesh sensitivities, introduced by the softening behavior of the matrix material, nonlocal integral-type material formulations are applied. The propagation of cracks at the interface between aggregates and mortar matrix is represented in a discrete way using a cohesive crack approach. The iterative solution procedure is stabilized using a new path following constraint within the framework of load-displacement-constraint methods which allows for an efficient representation of snap-back phenomena. In several examples, the influence of the randomly generated heterogeneous material structure on the stochastic scatter of the results is analyzed. Furthermore, the ability of mesoscale models to represent size effects is investigated. Mesoscale simulations require the discretization of the internal material structure. Compared to simulations on the macroscale, the numerical effort and the memory demand increases dramatically. Due to the complexity of the numerical model, mesoscale simulations are, in general, limited to small specimens. In this thesis, an adaptive heterogeneous multiscale approach is presented which allows for the incorporation of mesoscale models within nonlinear simulations of concrete structures. In heterogeneous multiscale models, only critical regions, i.e. regions in which damage develops, are resolved on the mesoscale, whereas undamaged or sparsely damage regions are modeled on the macroscale. A crucial point in simulations with heterogeneous multiscale models is the coupling of sub-domains discretized on different length scales. The sub-domains differ not only in the size of the finite elements but also in the constitutive description. In this thesis, different methods for the coupling of non-matching discretizations - constraint equations, the mortar method and the arlequin method - are investigated and the application to heterogeneous multiscale models is presented. Another important point is the detection of critical regions. An adaptive solution procedure allowing the transfer of macroscale sub-domains to the mesoscale is proposed. In this context, several indicators which trigger the model adaptation are introduced. Finally, the application of the proposed adaptive heterogeneous multiscale approach in nonlinear simulations of concrete structures is presented.

From a macroscopic point of view, failure within concrete structures is characterized by the initiation and propagation of cracks. In the first part of the thesis, a methodology for macroscopic crack growth simulations for concrete structures using a cohesive discrete crack approach based on the extended finite element method is introduced. Particular attention is turned to the investigation of criteria for crack initiation and crack growth. A drawback of the macroscopic simulation is that the real physical phenomena leading to the nonlinear behavior are only modeled phenomenologically. For concrete, the nonlinear behavior is characterized by the initiation of microcracks which coalesce into macroscopic cracks. In order to obtain a higher resolution of this failure zones, a mesoscale model for concrete is developed that models particles, mortar matrix and the interfacial transition zone (ITZ) explicitly. The essential features are a representation of particles using a prescribed grading curve, a material formulation based on a cohesive approach for the ITZ and a combined model with damage and plasticity for the mortar matrix. Compared to numerical simulations, the response of real structures exhibits a stochastic scatter. This is e.g. due to the intrinsic heterogeneities of the structure. For mesoscale models, these intrinsic heterogeneities are simulated by using a random distribution of particles and by a simulation of spatially variable material parameters using random fields. There are two major problems related to numerical simulations on the mesoscale. First of all, the material parameters for the constitutive description of the materials are often difficult to measure directly. In order to estimate material parameters from macroscopic experiments, a parameter identification procedure based on Bayesian neural networks is developed which is universally applicable to any parameter identification problem in numerical simulations based on experimental results. This approach offers information about the most probable set of material parameters based on experimental data and information about the accuracy of the estimate. Consequently, this approach can be used a priori to determine a set of experiments to be carried out in order to fit the parameters of a numerical model to experimental data. The second problem is the computational effort required for mesoscale simulations of a full macroscopic structure. For this purpose, a coupling between mesoscale and macroscale model is developed. Representative mesoscale simulations are used to train a metamodel that is finally used as a constitutive model in a macroscopic simulation. Special focus is placed on the ability of appropriately simulating unloading.

In recent years increasingly consideration has been given to the lifetime extension of existing structures. This is based on the fact that a growing percentage of civil infrastructure as well as buildings is threatened by obsolescence and that due to simple monetary reasons this can no longer be countered by simply re-building everything anew. Hence maintenance interventions are required which allow partial or complete structural rehabilitation. However, maintenance interventions have to be economically reasonable, that is, maintenance expenditures have to be outweighed by expected future benefits. Is this not the case, then indeed the structure is obsolete - at least in its current functional, economic, technical, or social configuration - and innovative alternatives have to be evaluated. An optimization formulation for planning maintenance interventions based on cost-benefit criteria is proposed herein. The underlying formulation is as follows: (a) between maintenance interventions structural deterioration is described as a random process; (b) maintenance interventions can take place anytime throughout lifetime and comprise the rehabilitation of all deterioration states above a certain minimum level; and (c) maintenance interventions are optimized by taking into account all expected life-cycle costs (construction, failure, inspection and state-dependent repair costs) as well as state- or time-dependent benefit rates. The optimization is performed by an evolutionary algorithm. The proposed approach also allows to determine optimal lifetimes and acceptable failure rates. Numerical examples demonstrate the importance of defining benefit rates explicitly. It is shown, that the optimal solution to maintenance interventions requires to take action before reaching the acceptable failure rate or the zero expected net benefit rate level. Deferring decisions with respect to maintenance not only results, in general, in higher losses, but also results in overly hazardous structures.

The importance of modern simulation methods in the mechanical analysis of heterogeneous solids is presented in detail. Thereby the problem is noted that even for small bodies the required high-resolution analysis reaches the limits of today's computational power, in terms of memory demand as well as acceptable computational effort. A further problem is that frequently the accuracy of geometrical modelling of heterogeneous bodies is inadequate. The present work introduces a systematic combination and adaption of grid-based methods for achieving an essentially higher resolution in the numerical analysis of heterogeneous solids. Grid-based methods are as well primely suited for developing efficient and numerically stable algorithms for flexible geometrical modeling. A key aspect is the uniform data management for a grid, which can be utilized to reduce the effort and complexity of almost all concerned methods. A new finite element program, called Mulgrido, was just developed to realize this concept consistently and to test the proposed methods. Several disadvantages which generally result from grid discretizations are selectively corrected by modified methods. The present work is structured into a geometrical model, a mechanical model and a numerical model. The geometrical model includes digital image-based modeling and in particular several methods for the theory-based generation of inclusion-matrix models. Essential contributions refer to variable shape, size distribution, separation checks and placement procedures of inclusions. The mechanical model prepares the fundamentals of continuum mechanics, homogenization and damage modeling for the following numerical methods. The first topic of the numerical model introduces to a special version of B-spline finite elements. These finite elements are entirely variable in the order k of B-splines. For homogeneous bodies this means that the approximation quality can arbitrarily be scaled. In addition, the multiphase finite element concept in combination with transition zones along material interfaces yields a valuable solution for heterogeneous bodies. As the formulation is element-based, the storage of a global stiffness matrix is superseded such that the memory demand can essentially be reduced. This is possible in combination with iterative solver methods which represent the second topic of the numerical model. Here, the focus lies on multigrid methods where the number of required operations to solve a linear equation system only increases linearly with problem size. Moreover, for badly conditioned problems quite an essential improvement is achieved by preconditioning. The third part of the numerical model discusses certain aspects of damage simulation which are closely related to the proposed grid discretization. The strong efficiency of the linear analysis can be maintained for damage simulation. This is achieved by a damage-controlled sequentially linear iteration scheme. Finally a study on the effective material behavior of heterogeneous bodies is presented. Especially the influence of inclusion shapes is examined. By means of altogether more than one hundred thousand random geometrical arrangements, the effective material behavior is statistically analyzed and assessed.