Classical and all-floating FETI methods for the simulation of arterial tissues

High-resolution and anatomically realistic computer models of biological soft tissues play a significant role in the understanding of the function of cardiovascular components in health and disease. However, the computational effort to handle fine grids to resolve the geometries as well as sophisticated tissue models is very challenging. One possibility to derive a strongly scalable parallel solution algorithm is to consider finite element tearing and interconnecting (FETI) methods. In this study we propose and investigate the application of FETI methods to simulate the elastic behavior of biological soft tissues. As one particular example we choose the artery which is - as most other biological tissues - characterized by anisotropic and nonlinear material properties. We compare two specific approaches of FETI methods, classical and all-floating, and investigate the numerical behavior of different preconditioning techniques. In comparison to classical FETI, the all-floating approach has not only advantages concerning the implementation but in many cases also concerning the convergence of the global iterative solution method. This behavior is illustrated with numerical examples. We present results of linear elastic simulations to show convergence rates, as expected from the theory, and results from the more sophisticated nonlinear case where we apply a well-known anisotropic model to the realistic geometry of an artery. Although the FETI methods have a great applicability on artery simulations we will also discuss some limitations concerning the dependence on material parameters.Comment: 29 page

On the initial estimate of interface forces in FETI methods

The Balanced Domain Decomposition (BDD) method and the Finite Element Tearing and Interconnecting (FETI) method are two commonly used non-overlapping domain decomposition methods. Due to strong theoretical and numerical similarities, these two methods are generally considered as being equivalently efficient. However, for some particular cases, such as for structures with strong heterogeneities, FETI requires a large number of iterations to compute the solution compared to BDD. In this paper, the origin of the bad efficiency of FETI in these particular cases is traced back to poor initial estimates of the interface stresses. To improve the estimation of interface forces a novel strategy for splitting interface forces between neighboring substructures is proposed. The additional computational cost incurred is not significant. This yields a new initialization for the FETI method and restores numerical efficiency which makes FETI comparable to BDD even for problems where FETI was performing poorly. Various simple test problems are presented to discuss the efficiency of the proposed strategy and to illustrate the so-obtained numerical equivalence between the BDD and FETI solvers

Isogeometric Simulation and Shape Optimization with Applications to Electrical Machines

Future e-mobility calls for efficient electrical machines. For different areas of operation, these machines have to satisfy certain desired properties that often depend on their design. Here we investigate the use of multipatch Isogeometric Analysis (IgA) for the simulation and shape optimization of the electrical machines. In order to get fast simulation and optimization results, we use non-overlapping domain decomposition (DD) methods to solve the large systems of algebraic equations arising from the IgA discretization of underlying partial differential equations. The DD is naturally related to the multipatch representation of the computational domain, and provides the framework for the parallelization of the DD solvers

Essential spectrum of local multi-trace boundary integral operators

Considering pure transmission scattering problems in piecewise constant media, we derive an exact analytic formula for the spectrum of the corresponding local multi-trace boundary integral operators in the case where the geometrical configuration does not involve any junction point and all wave numbers equal. We deduce from this the essential spectrum in the case where wave numbers vary. Numerical evidences of these theoretical results are also presented

Coupling of boundary element regions with the boundary element tearing and interconnecting method (BETI)

The boundary integral equation for elasticity is valid for a single domain consisting of homogeneous material properties. In the case of heterogeneity the consideration of different material properties is possible with a coupling of boundary element regions. Of course each region is again homogeneous. Another simulation application of multiple regions is the simulation of an industrial process, where different subdomains of a homogenous domain are treated differently due to a mechanical process. For instance, this is the case in tunnelling, where excavation is performed in a staged procedure. In the simulation of such an excavation process regions are deactivated step by step. As the material behaviour can be nonlinear an accurate simulation of such a staged process is a necessary requirement. Thus, the domain is decomposed into subregions which are coupled to neighbouring regions. There are different coupling strategies existing. In some of them stiffness matrices of subdomains are worked out which are the basis for the coupling and solution of the problem. A traditional method is the coupling of interface surfaces only [1]. In this method the stiffness matrix of a region is computed on the basis of the coupling surfaces (interfaces), whereas the coupling surface may be not identical to the complete surface of a subdomain and the size of the stiffness matrix is determined by the degrees of freedom of the coupling surface. In an application where the boundary conditions change (e.g. from interface to Neumann condition) from one calculation step to the other, the stiffness matrix has to be calculated new. A modern coupling technique is the Boundary Element Tearing and Interconnecting (BETI) method [2], similar to the method of Finite Element Tearing and Interconnecting (FETI) [3]. In this method the region stiffness matrix is worked out for the entire boundary of the region. The stiffness matrices of all regions remain the same during the whole analysis, even if the boundary conditions change during the simulation process. In setting up the equation system each subdomain is treated completely separated and independent from the others. Thus, a parallelisation of the computational work is ideally suited and implemented in the present computer code. In this work the theory of both mentioned coupling techniques are introduced briefly. The differences of both methods are worked out and advantages/disadvantages are shown and will be demonstrated. The accuracy of the results as well as the computational performance will be shown and compared based on a realistic simulation example