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- Finite-Elemente-Methode (3)
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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 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.

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 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.

This thesis addresses an adaptive higher-order method based on a Geometry Independent Field approximatTion(GIFT) of polynomial/rationals plines over hierarchical T-meshes(PHT/RHT-splines).
In isogeometric analysis, basis functions used for constructing geometric models in computer-aided design(CAD) are also employed to discretize the partial differential equations(PDEs) for numerical analysis. Non-uniform rational B-Splines(NURBS) are the most commonly used basis functions in CAD. However, they may not be ideal for numerical analysis where local refinement is required.
The alternative method GIFT deploys different splines for geometry and numerical analysis. NURBS are utilized for the geometry representation, while for the field solution, PHT/RHT-splines are used. PHT-splines not only inherit the useful properties of B-splines and NURBS, but also possess the capabilities of local refinement and hierarchical structure. The smooth basis function properties of PHT-splines make them suitable for analysis purposes. While most problems considered in isogeometric analysis can be solved efficiently when the solution is smooth, many non-trivial problems have rough solutions. For example, this can be caused by the presence of re-entrant corners in the domain. For such problems, a tensor-product basis (as in the case of NURBS) is less suitable for resolving the singularities that appear since refinement propagates throughout the computational domain. Hierarchical bases and local refinement (as in the case of PHT-splines) allow for a more efficient way to resolve these singularities by adding more degrees of freedom where they are necessary. In order to drive the adaptive refinement, an efficient recovery-based error estimator is proposed in this thesis. The estimator produces a recovery solution which is a more accurate approximation than the computed numerical solution. Several two- and three-dimensional numerical investigations with PHT-splines of higher order and continuity prove that the proposed method is capable of obtaining results with higher accuracy, better convergence, fewer degrees of freedom and less computational cost than NURBS for smooth solution problems. The adaptive GIFT method utilizing PHT-splines with the recovery-based error estimator is used for solutions with discontinuities or singularities where adaptive local refinement in particular domains of interest achieves higher accuracy with fewer degrees of freedom. This method also proves that it can handle complicated multi-patch domains for two- and three-dimensional problems outperforming uniform refinement in terms of degrees of freedom and computational cost.

Material properties play a critical role in durable products manufacturing. Estimation of the precise characteristics in different scales requires complex and expensive experimental measurements. Potentially, computational methods can provide a platform to determine the fundamental properties before the final experiment. Multi-scale computational modeling leads to the modeling of the various time, and length scales include nano, micro, meso, and macro scales. These scales can be modeled separately or in correlation with coarser scales. Depend on the interested scales modeling, the right selection of multi-scale methods leads to reliable results and affordable computational cost. The present dissertation deals with the problems in various length and time scales using computational methods include density functional theory (DFT), molecular mechanics (MM), molecular dynamics (MD), and finite element (FE) methods.
Physical and chemical interactions in lower scales determine the coarser scale properties. Particles interaction modeling and exploring fundamental properties are significant challenges of computational science. Downscale modelings need more computational effort due to a large number of interacted atoms/particles. To deal with this problem and bring up a fine-scale (nano) as a coarse-scale (macro) problem, we extended an atomic-continuum framework. The discrete atomic models solve as a continuum problem using the computationally efficient FE method. MM or force field method based on a set of assumptions approximates a solution on the atomic scale. In this method, atoms and bonds model as a harmonic oscillator with a system of mass and springs. The negative gradient of the potential energy equal to the forces on each atom. In this way, each bond's total potential energy includes bonded, and non-bonded energies are simulated as equivalent structural strain energies. Finally, the chemical nature of the atomic bond is modeled as a piezoelectric beam element that solves by the FE method.
Exploring novel materials with unique properties is a demand for various industrial applications. During the last decade, many two-dimensional (2D) materials have been synthesized and shown outstanding properties. Investigation of the probable defects during the formation/fabrication process and studying their strength under severe service life are the critical tasks to explore performance prospects. We studied various defects include nano crack, notch, and point vacancy (Stone-Wales defect) defects employing MD analysis. Classical MD has been used to simulate a considerable amount of molecules at micro-, and meso- scales. Pristine and defective nanosheet structures considered under the uniaxial tensile loading at various temperatures using open-source LAMMPS codes. The results were visualized with the open-source software of OVITO and VMD.
Quantum based first principle calculations have been conducting at electronic scales and known as the most accurate Ab initio methods. However, they are computationally expensive to apply for large systems. We used density functional theory (DFT) to estimate the mechanical and electrochemical response of the 2D materials. Many-body Schrödinger's equation describes the motion and interactions of the solid-state particles. Solid describes as a system of positive nuclei and negative electrons, all electromagnetically interacting with each other, where the wave function theory describes the quantum state of the set of particles. However, dealing with the 3N coordinates of the electrons, nuclei, and N coordinates of the electrons spin components makes the governing equation unsolvable for just a few interacted atoms. Some assumptions and theories like Born Oppenheimer and Hartree-Fock mean-field and Hohenberg-Kohn theories are needed to treat with this equation. First, Born Oppenheimer approximation reduces it to the only electronic coordinates. Then Kohn and Sham, based on Hartree-Fock and Hohenberg-Kohn theories, assumed an equivalent fictitious non-interacting electrons system as an electron density functional such that their ground state energies are equal to a set of interacting electrons. Exchange-correlation energy functionals are responsible for satisfying the equivalency between both systems. The exact form of the exchange-correlation functional is not known. However, there are widely used methods to derive functionals like local density approximation (LDA), Generalized gradient approximation (GGA), and hybrid functionals (e.g., B3LYP). In our study, DFT performed using VASP codes within the GGA/PBE approximation, and visualization/post-processing of the results realized via open-source software of VESTA.
The extensive DFT calculations are conducted 2D nanomaterials prospects as anode/cathode electrode materials for batteries. Metal-ion batteries' performance strongly depends on the design of novel electrode material. Two-dimensional (2D) materials have developed a remarkable interest in using as an electrode in battery cells due to their excellent properties. Desirable battery energy storage systems (BESS) must satisfy the high energy density, safe operation, and efficient production costs. Batteries have been using in electronic devices and provide a solution to the environmental issues and store the discontinuous energies generated from renewable wind or solar power plants. Therefore, exploring optimal electrode materials can improve storage capacity and charging/discharging rates, leading to the design of advanced batteries.
Our results in multiple scales highlight not only the proposed and employed methods' efficiencies but also promising prospect of recently synthesized nanomaterials and their applications as an anode material. In this way, first, a novel approach developed for the modeling of the 1D nanotube as a continuum piezoelectric beam element. The results converged and matched closely with those from experiments and other more complex models. Then mechanical properties of nanosheets estimated and the failure mechanisms results provide a useful guide for further use in prospect applications. Our results indicated a comprehensive and useful vision concerning the mechanical properties of nanosheets with/without defects. Finally, mechanical and electrochemical properties of the several 2D nanomaterials are explored for the first time—their application performance as an anode material illustrates high potentials in manufacturing super-stretchable and ultrahigh-capacity battery energy storage systems (BESS). Our results exhibited better performance in comparison to the available commercial anode materials.