ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar
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2022,9
Material failure can be tackled by so-called nonlocal models, which introduce an intrinsic length scale into the formulation and, in the case of material failure, restore the well-posedness of the underlying boundary value problem or initial boundary value problem. Among nonlocal models, peridynamics (PD) has attracted a lot of attention as it allows the natural transition from continuum to discontinue and thus allows modeling of discrete cracks without the need to describe and track the crack topology, which has been a major obstacle in traditional discrete crack approaches. This is achieved by replacing the divergence of the Cauchy stress tensor through an integral over so-called bond forces, which account for the interaction of particles. A quasi-continuum approach is then used to calibrate the material parameters of the bond forces, i.e., equating the PD energy with the energy of a continuum. One major issue for the application of PD to general complex problems is that they are limited to fairly simple material behavior and pure mechanical problems based on explicit time integration. PD has been extended to other applications but losing simultaneously its simplicity and ease in modeling material failure. Furthermore, conventional PD suffers from instability and hourglass modes that require stabilization. It also requires the use of constant horizon sizes, which drastically reduces its computational efficiency. The latter issue was resolved by the so-called dual-horizon peridynamics (DH-PD) formulation and the introduction of the duality of horizons.
Within the nonlocal operator method (NOM), the concept of nonlocality is further extended and can be considered a generalization of DH-PD. Combined with the energy functionals of various physical models, the nonlocal forms based on the dual-support concept can be derived. In addition, the variation of the energy functional allows implicit formulations of the nonlocal theory. While traditional integral equations are formulated in an integral domain, the dual-support approaches are based on dual integral domains. One prominent feature of NOM is its compatibility with variational and weighted residual methods. The NOM yields a direct numerical implementation based on the weighted residual method for many physical problems without the need for shape functions. Only the definition of the energy or boundary value problem is needed to drastically facilitate the implementation. The nonlocal operator plays an equivalent role to the derivatives of the shape functions in meshless methods and finite element methods (FEM). Based on the variational principle, the residual and the tangent stiffness matrix can be obtained with ease by a series of matrix multiplications. In addition, NOM can be used to derive many nonlocal models in strong form.
The principal contributions of this dissertation are the implementation and application of NOM, and also the development of approaches for dealing with fractures within the NOM, mostly for dynamic fractures. The primary coverage and results of the dissertation are as follows:
-The first/higher-order implicit NOM and explicit NOM, including a detailed description of the implementation, are presented. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combining with the method of weighted residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. For the sake of conciseness, the implementation in this chapter is focused on linear elastic solids only, though the NOM can handle more complex nonlinear problems. An explicit nonlocal operator method for the dynamic analysis of elasticity solid problems is also presented. The explicit NOM avoids the calculation of the tangent stiffness matrix as in the implicit NOM model. The explicit scheme comprises the Verlet-velocity algorithm. The NOM can be very flexible and efficient for solving partial differential equations (PDEs). It's also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Several numerical examples are presented to show the capabilities of this method.
-A nonlocal operator method for the dynamic analysis of (thin) Kirchhoff plates is proposed. The nonlocal Hessian operator is derived from a second-order Taylor series expansion. NOM is higher-order continuous, which is exploited for thin plate analysis that requires $C^1$ continuity. The nonlocal dynamic governing formulation and operator energy functional for Kirchhoff plates are derived from a variational principle. The Verlet-velocity algorithm is used for time discretization. After confirming the accuracy of the nonlocal Hessian operator, several numerical examples are simulated by the nonlocal dynamic Kirchhoff plate formulation.
-A nonlocal fracture modeling is developed and applied to the simulation of quasi-static and dynamic fractures using the NOM. The phase field's nonlocal weak and associated strong forms are derived from a variational principle. The NOM requires only the definition of energy. We present both a nonlocal implicit phase field model and a nonlocal explicit phase field model for fracture; the first approach is better suited for quasi-static fracture problems, while the key application of the latter one is dynamic fracture. To demonstrate the performance of the underlying approach, several benchmark examples for quasi-static and dynamic fracture are solved.
2022,8
Finite Element Simulations of dynamically excited structures are mainly influenced by the mass, stiffness, and damping properties of the system, as well as external loads. The prediction quality of dynamic simulations of vibration-sensitive components depends significantly on the use of appropriate damping models. Damping phenomena have a decisive influence on the vibration amplitude and the frequencies of the vibrating structure. However, developing realistic damping models is challenging due to the multiple sources that cause energy dissipation, such as material damping, different types of friction, or various interactions with the environment.
This thesis focuses on thermoelastic damping, which is the main cause of material damping in homogeneous materials. The effect is caused by temperature changes due to mechanical strains. In vibrating structures, temperature gradients arise in adjacent tension and compression areas. Depending on the vibration frequency, they result in heat flows, leading to increased entropy and the irreversible transformation of mechanical energy into thermal energy.
The central objective of this thesis is the development of efficient simulation methods to incorporate thermoelastic damping in finite element analyses based on modal superposition. The thermoelastic loss factor is derived from the structure's mechanical mode shapes and eigenfrequencies. In subsequent analyses that are performed in the time and frequency domain, it is applied as modal damping.
Two approaches are developed to determine the thermoelastic loss in thin-walled plate structures, as well as three-dimensional solid structures. The realistic representation of the dissipation effects is verified by comparing the simulation results with experimentally determined data. Therefore, an experimental setup is developed to measure material damping, excluding other sources of energy dissipation.
The three-dimensional solid approach is based on the determination of the generated entropy and therefore the generated heat per vibration cycle, which is a measure for thermoelastic loss in relation to the total strain energy. For thin plate structures, the amount of bending energy in a modal deformation is calculated and summarized in the so-called Modal Bending Factor (MBF). The highest amount of thermoelastic loss occurs in the state of pure bending. Therefore, the MBF enables a quantitative classification of the mode shapes concerning the thermoelastic damping potential.
The results of the developed simulations are in good agreement with the experimental results and are appropriate to predict thermoelastic loss factors. Both approaches are based on modal superposition with the advantage of a high computational efficiency. Overall, the modeling of thermoelastic damping represents an important component in a comprehensive damping model, which is necessary to perform realistic simulations of vibration processes.
2022,3
In recent years, lightweight materials, such as polymer composite materials (PNCs) have been studied and developed due to their excellent physical and chemical properties. Structures composed of these composite materials are widely used in aerospace engineering structures, automotive components, and electrical devices. The excellent and outstanding mechanical, thermal, and electrical properties of Carbon nanotube (CNT) make it an ideal filler to strengthen polymer materials’ comparable properties. The heat transfer of composite materials has very promising engineering applications in many fields, especially in electronic devices and energy storage equipment. It is essential in high-energy density systems since electronic components need heat dissipation functionality. Or in other words, in electronic devices the generated heat should ideally be dissipated by light and small heat sinks.
Polymeric composites consist of fillers embedded in a polymer matrix, the first ones will significantly affect the overall (macroscopic) performance of the material. There are many common carbon-based fillers such as single-walled carbon nanotubes (SWCNT), multi-walled carbon nanotubes (MWCNT), carbon nanobuds (CNB), fullerene, and graphene. Additives inside the matrix have become a popular subject for researchers. Some extraordinary characters, such as high-performance load, lightweight design, excellent chemical resistance, easy processing, and heat transfer, make the design of polymeric nanotube composites (PNCs) flexible. Due to the reinforcing effects with different fillers on composite materials, it has a higher degree of freedom and can be designed for the structure according to specific applications’ needs. As already stated, our research focus will be on SWCNT enhanced PNCs. Since experiments are timeconsuming, sometimes expensive and cannot shed light into phenomena taking place for instance at the interfaces/interphases of composites, they are often complemented through theoretical and computational analysis.
While most studies are based on deterministic approaches, there is a comparatively lower number of stochastic methods accounting for uncertainties in the input parameters. In deterministic models, the output of the model is fully determined by the parameter values and the initial conditions. However, uncertainties in the input parameters such as aspect ratio, volume fraction, thermal properties of fiber and matrix need to be taken into account for reliable predictions. In this research, a stochastic multiscale method is provided to study the influence of numerous uncertain input parameters on the thermal conductivity of the composite. Therefore, a hierarchical multi-scale method based on computational homogenization is presented in to predict the macroscopic thermal conductivity based on the fine-scale structure. In order to study the inner mechanism, we use the finite element method and employ surrogate models to conduct a Global Sensitivity Analysis (GSA). The SA is performed in order to quantify the influence of the conductivity of the fiber, matrix, Kapitza resistance, volume fraction and aspect ratio on the macroscopic conductivity. Therefore, we compute first-order and total-effect sensitivity indices with different surrogate models.
As stochastic multiscale models are computational expensive, surrogate approaches are commonly exploited. With the emergence of high performance computing and artificial intelligence, machine learning has become a popular modeling tool for numerous applications. Machine learning (ML) is commonly used in regression and maps data through specific rules with algorithms to build input and output models. They are particularly useful for nonlinear input-output relationships when sufficient data is available. ML has also been used in the design of new materials and multiscale analysis. For instance, Artificial neural networks and integrated learning seem to be ideally for such a task. They can theoretically simulate any non-linear relationship through the connection of neurons. Mapping relationships are employed to carry out data-driven simulations of inputs and outputs in stochastic modeling.
This research aims to develop a stochastic multi-scale computational models of PNCs in heat transfer. Multi-scale stochastic modeling with uncertainty analysis and machine learning methods consist of the following components:
-Uncertainty Analysis. A surrogate based global sensitivity analysis is coupled with a hierarchical multi-scale method employing computational homogenization. The effect of the conductivity of the fibers and the matrix, the Kapitza resistance, volume fraction and aspect ratio on the ’macroscopic’ conductivity of the composite is systematically studied. All selected surrogate models yield consistently the conclusions that the most influential input parameters are the aspect ratio followed by the volume fraction. The Kapitza Resistance has no significant effect on the thermal conductivity of the PNCs. The most accurate surrogate model in terms of the R2 value is the moving least square (MLS).
-Hybrid Machine Learning Algorithms. A combination of artificial neural network (ANN) and particle swarm optimization (PSO) is applied to estimate the relationship between variable input and output parameters. The ANN is used for modeling the composite while PSO improves the prediction performance through an optimized global minimum search. The thermal conductivity of the fibers and the matrix, the kapitza resistance, volume fraction and aspect ratio are selected as input parameters. The output is the macroscopic (homogenized) thermal conductivity of the composite. The results show that the PSO significantly improves the predictive ability of this hybrid intelligent algorithm, which outperforms traditional neural networks.
-Stochastic Integrated Machine Learning. A stochastic integrated machine learning based multiscale approach for the prediction of the macroscopic thermal conductivity in PNCs is developed. Seven types of machine learning models are exploited in this research, namely Multivariate Adaptive Regression Splines (MARS), Support Vector Machine (SVM), Regression Tree (RT), Bagging Tree (Bag), Random Forest (RF), Gradient Boosting Machine (GBM) and Cubist. They are used as components of stochastic modeling to construct the relationship between the variable of the inputs’ uncertainty and the macroscopic thermal conductivity of PNCs. Particle Swarm Optimization (PSO) is used for hyper-parameter tuning to find the global optimal values leading to a significant reduction in the computational cost. The advantages and disadvantages of various methods are also analyzed in terms of computing time and model complexity to finally give a recommendation for the applicability of different models.
2022,1
Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDEs), which was introduced with the aim of integrating finite element analysis with computer-aided design systems. The main idea of the method is to use the same spline basis functions which describe the geometry in CAD systems for the approximation of solution fields in the finite element method (FEM). Originally, NURBS which is a standard technology employed in CAD systems was adopted as basis functions in IGA but there were several variants of IGA using other technologies such as T-splines, PHT splines, and subdivision surfaces as basis functions. In general, IGA offers two key advantages over classical FEM: (i) by describing the CAD geometry exactly using smooth, high-order spline functions, the mesh generation process is simplified and the interoperability between CAD and FEM is improved, (ii) IGA can be viewed as a high-order finite element method which offers basis functions with high inter-element continuity and therefore can provide a primal variational formulation of high-order PDEs in a straightforward fashion. The main goal of this thesis is to further advance isogeometric analysis by exploiting these major advantages, namely precise geometric modeling and the use of smooth high-order splines as basis functions, and develop robust computational methods for problems with complex geometry and/or complex multi-physics.
As the first contribution of this thesis, we leverage the precise geometric modeling of isogeometric analysis and propose a new method for its coupling with meshfree discretizations. We exploit the strengths of both methods by using IGA to provide a smooth, geometrically-exact surface discretization of the problem domain boundary, while the Reproducing Kernel Particle Method (RKPM) discretization is used to provide the volumetric discretization of the domain interior. The coupling strategy is based upon the higher-order consistency or reproducing conditions that are directly imposed in the physical domain. The resulting coupled method enjoys several favorable features: (i) it preserves the geometric exactness of IGA, (ii) it circumvents the need for global volumetric parameterization of the problem domain, (iii) it achieves arbitrary-order approximation accuracy while preserving higher-order smoothness of the discretization. Several numerical examples are solved to show the optimal convergence properties of the coupled IGA–RKPM formulation, and to demonstrate its effectiveness in constructing volumetric discretizations for complex-geometry objects.
As for the next contribution, we exploit the use of smooth, high-order spline basis functions in IGA to solve high-order surface PDEs governing the morphological evolution of vesicles. These governing equations are often consisted of geometric PDEs, high-order PDEs on stationary or evolving surfaces, or a combination of them. We propose an isogeometric formulation for solving these PDEs. In the context of geometric PDEs, we consider phase-field approximations of mean curvature flow and Willmore flow problems and numerically study the convergence behavior of isogeometric analysis for these problems. As a model problem for high-order PDEs on stationary surfaces, we consider the Cahn–Hilliard equation on a sphere, where the surface is modeled using a phase-field approach. As for the high-order PDEs on evolving surfaces, a phase-field model of a deforming multi-component vesicle, which consists of two fourth-order nonlinear PDEs, is solved using the isogeometric analysis in a primal variational framework. Through several numerical examples in 2D, 3D and axisymmetric 3D settings, we show the robustness of IGA for solving the considered phase-field models.
Finally, we present a monolithic, implicit formulation based on isogeometric analysis and generalized-alpha time integration for simulating hydrodynamics of vesicles according to a phase-field model. Compared to earlier works, the number of equations of the phase-field model which need to be solved is reduced by leveraging high continuity of NURBS functions, and the algorithm is extended to 3D settings. We use residual-based variational multi-scale method (RBVMS) for solving Navier–Stokes equations, while the rest of PDEs in the phase-field model are treated using a standard Galerkin-based IGA. We introduce the resistive immersed surface (RIS) method into the formulation which can be employed for an implicit description of complex geometries using a diffuse-interface approach. The implementation highlights the robustness of the RBVMS method for Navier–Stokes equations of incompressible flows with non-trivial localized forcing terms including bending and tension forces of the vesicle. The potential of the phase-field model and isogeometric analysis for accurate simulation of a variety of fluid-vesicle interaction problems in 2D and 3D is demonstrated.