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#### Keywords

- FEM (1)
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- Mesh quality (1)
- Model quality, Model error estimation, Kinematical model, Geometric non-linearity, Finite Element method (1)
- Modellqualität, Modellfehlerschätzer, Geometrisch nicht-lineare Berechnung, Kinematik Modell, Finite Elemente Methode (1)
- NURBS (1)
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#### Year of publication

- 2012 (3) (remove)

Methods for model quality assessment are aiming to find the most appropriate model with respect to accuracy and computational effort for a structural system under investigation. Model error estimation techniques can be applied for this purpose when kinematical models are investigated. They are counted among the class of white box models, which means that the model hierarchy and therewith the best model is known. This thesis gives an overview of discretisation error estimators. Deduced from these, methods for model error estimation are presented. Their general goal is to make a prediction of the inaccuracies that are introduced using the simpler model without knowing the solution of a more complex model. This information can be used to steer an adaptive process. Techniques for linear and non-linear problems as well as global and goal-oriented errors are introduced. The estimation of the error in local quantities is realised by solving a dual problem, which serves as a weight for the primal error. So far, such techniques have mainly been applied in
material modelling and for dimensional adaptivity. Within the scope of this thesis, available model error estimators are adapted for an application to kinematical models. Their applicability is tested regarding the question of whether a geometrical non-linear calculation is necessary or not. The analysis is limited to non-linear estimators due to the structure of the underlying differential equations. These methods often involve simplification, e.g linearisations. It is investigated to which extent such assumptions lead to meaningful results, when applied to kinematical models.

Increasingly powerful hard- and software allows for the numerical simulation of complex physical phenomena with high levels of detail. In light of this development the definition of numerical models for the Finite Element Method (FEM) has become the bottleneck in the simulation process. Characteristic features of the model generation are large manual efforts and a de-coupling of geometric and numerical model. In the highly probable case of design revisions all steps of model preprocessing and mesh generation have to be repeated. This includes the idealization and approximation of a geometric model as well as the definition of boundary conditions and model parameters. Design variants leading to more resource-efficient structures might hence be disregarded due to limited budgets and constrained time frames.
A potential solution to above problem is given with the concept of Isogeometric Analysis (IGA). Core idea of this method is to directly employ a geometric model for numerical simulations, which allows to circumvent model transformations and the accompanying data losses. Basis for this method are geometric models described in terms of Non-uniform rational B-Splines (NURBS). This class of piecewise continuous rational polynomial functions is ubiquitous in computer graphics and Computer-Aided Design (CAD). It allows the description of a wide range of geometries using a compact mathematical representation. The shape of an object thereby results from the interpolation of a set of control points by means of the NURBS functions, allowing efficient representations for curves, surfaces and solid bodies alike. Existing software applications, however, only support the modeling and manipulation of the former two. The description of three-dimensional solid bodies consequently requires significant manual effort, thus essentially forbidding the setup of complex models.
This thesis proposes a procedural approach for the generation of volumetric NURBS models. That is, a model is not described in terms of its data structures but as a sequence of modeling operations applied to a simple initial shape. In a sense this describes the "evolution" of the geometric model under the sequence of operations. In order to adapt this concept to NURBS geometries, only a compact set of commands is necessary which, in turn, can be adapted from existing algorithms. A model then can be treated in terms of interpretable model parameters. This leads to an abstraction from its data structures and model variants can be set up by variation of the governing parameters.
The proposed concept complements existing template modeling approaches: templates can not only be defined in terms of modeling commands but can also serve as input geometry for said operations. Such templates, arranged in a nested hierarchy, provide an elegant model representation. They offer adaptivity on each tier of the model hierarchy and allow to create complex models from only few model parameters. This is demonstrated for volumetric fluid domains used in the simulation of vertical-axis wind turbines. Starting from a template representation of airfoil cross-sections, the complete "negative space" around the rotor blades can be described by a small set of model parameters, and model variants can be set up in a fraction of a second.
NURBS models offer a high geometric flexibility, allowing to represent a given shape in different ways. Different model instances can exhibit varying suitability for numerical analyses. For their assessment, Finite Element mesh quality metrics are regarded. The considered metrics are based on purely geometric criteria and allow to identify model degenerations commonly used to achieve certain geometric features. They can be used to decide upon model adaptions and provide a measure for their efficacy. Unfortunately, they do not reveal a relation between mesh distortion and ill-conditioning of the equation systems resulting from the numerical model.

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.