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The aim of this study is controlling of spurious oscillations developing around discontinuous solutions of both linear and non-linear wave equations or hyperbolic partial differential equations (PDEs). The equations include both first-order and second-order (wave) hyperbolic systems. In these systems even smooth initial conditions, or smoothly varying source (load) terms could lead to discontinuous propagating solutions (fronts). For the first order hyperbolic PDEs, the concept of central high resolution schemes is integrated with the multiresolution-based adaptation to capture properly both discontinuous propagating fronts and effects of fine-scale responses on those of larger scales in the multiscale manner. This integration leads to using central high resolution schemes on non-uniform grids; however, such simulation is unstable, as the central schemes are originally developed to work properly on uniform cells/grids. Hence, the main concern is stable collaboration of central schemes and multiresoltion-based cell adapters. Regarding central schemes, the considered approaches are: 1) Second order central and central-upwind schemes; 2) Third order central schemes; 3) Third and fourth order central weighted non-oscillatory schemes (central-WENO or CWENO); 4) Piece-wise parabolic methods (PPMs) obtained with two different local stencils. For these methods, corresponding (nonlinear) stability conditions are studied and modified, as well. Based on these stability conditions several limiters are modified/developed as follows: 1) Several second-order limiters with total variation diminishing (TVD) feature, 2) Second-order uniformly high order accurate non-oscillatory (UNO) limiters, 3) Two third-order nonlinear scaling limiters, 4) Two new limiters for PPMs. Numerical results show that adaptive solvers lead to cost-effective computations (e.g., in some 1-D problems, number of adapted grid points are less than 200 points during simulations, while in the uniform-grid case, to have the same accuracy, using of 2049 points is essential). Also, in some cases, it is confirmed that fine scale responses have considerable effects on higher scales.
In numerical simulation of nonlinear first order hyperbolic systems, the two main concerns are: convergence and uniqueness. The former is important due to developing of the spurious oscillations, the numerical dispersion and the numerical dissipation. Convergence in a numerical solution does not guarantee that it is the physical/real one (the uniqueness feature). Indeed, a nonlinear systems can converge to several numerical results (which mathematically all of them are true). In this work, the convergence and uniqueness are directly studied on non-uniform grids/cells by the concepts of local numerical truncation error and numerical entropy production, respectively. Also, both of these concepts have been used for cell/grid adaptations. So, the performance of these concepts is also compared by the multiresolution-based method. Several 1-D and 2-D numerical examples are examined to confirm the efficiency of the adaptive solver. Examples involve problems with convex and non-convex fluxes. In the latter case, due to developing of complex waves, proper capturing of real answers needs more attention. For this purpose, using of method-adaptation seems to be essential (in parallel to the cell/grid adaptation). This new type of adaptation is also performed in the framework of the multiresolution analysis.
Regarding second order hyperbolic PDEs (mechanical waves), the regularization concept is used to cure artificial (numerical) oscillation effects, especially for high-gradient or discontinuous solutions. There, oscillations are removed by the regularization concept acting as a post-processor. Simulations will be performed directly on the second-order form of wave equations. It should be mentioned that it is possible to rewrite second order wave equations as a system of first-order waves, and then simulated the new system by high resolution schemes. However, this approach ends to increasing of variable numbers (especially for 3D problems).
The numerical discretization is performed by the compact finite difference (FD) formulation with desire feature; e.g., methods with spectral-like or optimized-error properties. These FD methods are developed to handle high frequency waves (such as waves near earthquake sources). The performance of several regularization approaches is studied (both theoretically and numerically); at last, a proper regularization approach controlling the Gibbs phenomenon is recommended.
At the end, some numerical results are provided to confirm efficiency of numerical solvers enhanced by the regularization concept. In this part, shock-like responses due to local and abrupt changing of physical properties, and also stress wave propagation in stochastic-like domains are studied.

The p-Laplace equation is a nonlinear generalization of the well-known Laplace equation. It is often used as a model problem for special types of nonlinearities, and therefore it can be seen as a bridge between very general nonlinear equations and the linear Laplace equation, too. It appears in many problems for instance in the theory of non-Newtonian fluids and fluid dynamics or in rockfill dam problems, as well as in special problems of image restoration and image processing.
The aim of this thesis is to solve the p-Laplace equation for 1 < p < 2, as well as for 2 < p < 3 and to find strong solutions in the framework of Clifford analysis. The idea is to apply a hypercomplex integral operator and special function theoretic methods to transform the p-Laplace equation into a p-Dirac equation. We consider boundary value problems for the p-Laplace equation and transfer them to boundary value problems for a p-Dirac equation. These equations will be solved iteratively by applying Banach’s fixed-point principle. Applying operator-theoretical methods for the p-Dirac equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved.
In addition, using a finite difference approach on a uniform lattice in the plane, the fundamental solution of the Cauchy-Riemann operator and its adjoint based on the fundamental solution of the Laplacian will be calculated. Besides, we define gener- alized discrete Teodorescu transform operators, which are right-inverse to the discrete Cauchy-Riemann operator and its adjoint in the plane. Furthermore, a new formula for generalized discrete boundary operators (analogues of the Cauchy integral operator) will be considered. Based on these operators a new version of discrete Borel-Pompeiu formula is formulated and proved.
This is the basis for an operator calculus that will be applied to the numerical solution of the p-Dirac equation. Finally, numerical results will be presented showing advantages and problems of this approach.

The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part.
First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of “isoparametric”, for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images.
Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids.
Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1
continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems.

This thesis presents two new methods in finite elements and isogeometric analysis for structural analysis. The first method proposes an alternative alpha finite element method using triangular elements. In this method, the piecewise constant strain field of linear triangular finite element method models is enhanced by additional strain terms with an adjustable parameter a, which results in an effectively softer stiffness formulation compared to a linear triangular element. In order to avoid the transverse shear locking of Reissner-Mindlin plates analysis the alpha finite element method is coupled with a discrete shear gap technique for triangular elements to significantly improve the accuracy of the standard triangular finite elements.
The basic idea behind this element formulation is to approximate displacements and rotations as in the standard finite element method, but to construct the bending, geometrical and shear strains using node-based smoothing domains. Several numerical examples are presented and show that the alpha FEM gives a good agreement compared to several other methods in the literature.
Second method, isogeometric analysis based on rational splines over hierarchical T-meshes (RHT-splines) is proposed. The RHT-splines are a generalization of Non-Uniform Rational B-splines (NURBS) over hierarchical T-meshes, which is a piecewise bicubic polynomial over a hierarchical
T-mesh. The RHT-splines basis functions not only inherit all the properties of NURBS such as non-negativity, local support and partition of unity but also more importantly as the capability of joining geometric objects without gaps, preserving higher order continuity everywhere and allow local refinement and adaptivity. In order to drive the adaptive refinement, an efficient recovery-based error estimator is employed. For this problem an imaginary surface is defined. The imaginary surface is basically constructed by RHT-splines basis functions which is used for approximation and interpolation functions as well as the construction of the recovered stress components. Numerical investigations prove that the proposed method is capable to obtain results with higher accuracy and convergence rate than NURBS results.

In the last decades, Finite Element Method has become the main method in statics and dynamics analysis in engineering practice. For current problems, this method provides a faster, more flexible solution than the analytic approach. Prognoses of complex engineer problems that used to be almost impossible to solve are now feasible.
Although the finite element method is a robust tool, it leads to new questions about engineering solutions. Among these new problems, it is possible to divide into two major groups: the first group is regarding computer performance; the second one is related to understanding the digital solution.
Simultaneously with the development of the finite element method for numerical solutions, a theory between beam theory and shell theory was developed: Generalized Beam Theory, GBT. This theory has not only a systematic and analytical clear presentation of complicated structural problems, but also a compact and elegant calculation approach that can improve computer performance.
Regrettably, GBT was not internationally known since the most publications of this theory were written in German, especially in the first years. Only in recent years, GBT has gradually become a fertile research topic, with developments from linear to non-linear analysis.
Another reason for the misuse of GBT is the isolated application of the theory. Although recently researches apply finite element method to solve the GBT's problems numerically, the coupling between finite elements of GBT and other theories (shell, solid, etc) is not the subject of previous research. Thus, the main goal of this dissertation is the coupling between GBT and shell/membrane elements. Consequently, one achieves the benefits of both sides: the versatility of shell elements with the high performance of GBT elements.
Based on the assumptions of GBT, this dissertation presents how the separation of variables leads to two calculation's domains of a beam structure: a cross-section modal analysis and the longitudinal amplification axis. Therefore, there is the possibility of applying the finite element method not only in the cross-section analysis, but also the development for an exact GBT's finite element in the longitudinal direction.
For the cross-section analysis, this dissertation presents the solution of the quadratic eigenvalue problem with an original separation between plate and membrane mechanism. Subsequently, one obtains a clearer representation of the deformation mode, as well as a reduced quadratic eigenvalue problem.
Concerning the longitudinal direction, this dissertation develops the novel exact elements, based on hyperbolic and trigonometric shape functions. Although these functions do not have trivial expressions, they provide a recursive procedure that allows periodic derivatives to systematise the development of stiffness matrices. Also, these shape functions enable a single-element discretisation of the beam structure and ensure a smooth stress field.
From these developments, this dissertation achieves the formulation of its primary objective: the connection of GBT and shell elements in a mixed model. Based on the displacement field, it is possible to define the coupling equations applied in the master-slave method. Therefore, one can model the structural connections and joints with finite shell elements and the structural beams and columns with GBT finite element.
As a side effect, the coupling equations limit the displacement field of the shell elements under the assumptions of GBT, in particular in the neighbourhood of the coupling cross-section.
Although these side effects are almost unnoticeable in linear analysis, they lead to cumulative errors in non-linear analysis. Therefore, this thesis finishes with the evaluation of the mixed GBT-shell models in non-linear analysis.