Alexander Düster

PERFORMANCE OF DIFFERENT INTEGRATION SCHEMES IN FACING DISCONTINUITIES IN THE FINITE CELL METHOD

In many extended versions of the finite element method (FEM) the mesh does not conform to the physical domain. Therefore, discontinuity of variables is expected when some elements are cut by the boundary. Thus, the integrands are not continuous over the whole integration domain. Apparently, none of the well developed integration schemes such as Gauss quadrature can be used readily. This paper investigates several modifications of the Gauss quadrature to capture the discontinuity within an element and to perform a more precise integration. The extended method used here is the finite cell method (FCM), an extension of a high-order approximation space with the aim of simple meshing. Several examples are included to evaluate different modifications.

The p-version of the finite element method compared to an adaptive h-version for the deformation theory of plasticity

A p-version of the finite element method is applied to the deformation theory of plasticity and the results are compared to a state-of-the-art adaptive h-version. It is demonstrated that even for nonlinear elliptic problems the p-version is a very efficient discretization strategy.

Accelerated staggered coupling schemes for problems of thermoelasticity at finite strains

This paper introduces a fully implicit partitioned coupling scheme for problems of thermoelasticity at finite strains utilizing the p-version of the finite element method. The mechanical and the thermal fields are partitioned into symmetric subproblems where algorithmic decoupling has been obtained by means of an isothermal operator-split. Numerical relaxation methods have been implemented to accelerate the convergence of the algorithm. Such methods are well-known from coupled fluid-structure interaction problems leading to highly efficient algorithms. Having studied the influence of three different strategies: polynomial prediction methods, numerical relaxation with constant relaxation coefficients, its dynamic variant with a residual based relaxation coefficient and a variant of a reduced order model - quasi-Newton method, we present several numerical simulations of quasi-static problems investigating the performance of accelerated coupling schemes.

Non-standard bone simulation: interactive numerical analysis by computational steering

Numerous numerical methods have been developed in an effort to accurately predict stresses in bones. The largest group are variants of the h-version of the finite element method (h-FEM), where low order Ansatz functions are used. By contrast, we3 investigate a combination of high order FEM and a fictitious domain approach, the finite cell method (FCM). While the FCM has been verified and validated in previous publications, this article proposes methods on how the FCM can be made computationally efficient to the extent that it can be used for patient specific, interactive bone simulations. This approach is called computational steering and allows to change input parameters like the position of an implant, material or loads and leads to an almost instantaneous change in the output (stress lines, deformations). This direct feedback gives the user an immediate impression of the impact of his actions to an extent which, otherwise, is hard to obtain by the use of classical non interactive computations. Specifically, we investigate an application to pre-surgical planning of a total hip replacement where it is desirable to select an optimal implant for a specific patient. Herein, optimal is meant in the sense that the expected post-operative stress distribution in the bone closely resembles that before the operation.

Theoretical and Numerical Investigation of the Finite Cell Method

We present a detailed analysis of the convergence properties of the finite cell method which is a fictitious domain approach based on high order finite elements. It is proved that exponential type of convergence can be obtained by the finite cell method for Laplace and Lamé problems in one, two as well as three dimensions. Several numerical examples in one and two dimensions including a well-known benchmark problem from linear elasticity confirm the results of the mathematical analysis of the finite cell method.

FCMLab: A finite cell research toolbox for MATLAB

The recently introduced Finite Cell Method combines the fictitious domain idea with the benefits of high-order finite elements. Although previous publications demonstrated the method’s excellent applicability in various contexts, the implementation of a three-dimensional Finite Cell code is challenging. To lower the entry barrier, this work introduces the object-oriented MATLAB toolbox FCMLab allowing for an easy start into this research field and for rapid prototyping of new algorithmic ideas. The paper reviews the essentials of the methods applied and explains in detail the class structure of the framework. Furthermore, the usage of the toolbox is discussed by means of different two- and three-dimensional examples demonstrating all important features of FCMLab (http://fcmlab.cie.bgu.tum.de/).

Thin Solids for Fluid-Structure Interaction

In this contribution the use of hexahedral elements for the structural simulation in a fluid structure interaction framework is presented, resulting in a consistent kinematic and geometric description of the solid. In order to compensate the additional numerical effort of the three-dimensional approach, an anisotropic p-adaptive method for linear elastodynamic problems is proposed, resulting in a clearly higher efficiency and higher convergence rates than uniform p-extensions. Special emphasis is placed on the accurate transfer of loads considering the fluid discretization for computation of the surface load integrals. For a coupling with a cartesian grid based Lattice Boltzmann code it was found that oscillations in the interface tractions may excite higher structural modes possibly leading to a non-stable coupling behavior. A first remedy to this problem was a linear modal analysis of the structure, thus allowing to control the number of modes to be considered without disregarding bidirectional fluid structure interactions. Preliminary results are presented for the FSI benchmark configuration proposed in this book.

A Numerical Investigation of High-Order Finite Elements for Problems of Elastoplasticity

A high order finite element approach is applied to elastoplastic problems in two as well as in three dimensions. The element formulations are based on quadrilaterals and hexahedrals, taking advantage of the blending function method in order to accurately represent the geometry. A comparison of h- and p-extensions is drawn and it is shown that thin-walled structures commonly being analysed by dimensionally reduced elements may be consistently discretized by high order hexahedral elements leading to reliable and efficient computations even in case of physically nonlinear problems.

Simulation of Lamb waves using the spectral cell method

Today a steadily growing interest in on-line monitoring of structures is seen. Commonly referred to as structural health monitoring (SHM), the basic idea of this technique is to decrease the maintenance costs based on a continuous flow of information concerning the state of the structure. With respect to the aeronautic industry increasing the service time of airplanes is another important goal. A popular approach to SHM is to be seen in ultrasonic guided wave based monitoring systems. Since one focus is on typical lightweight materials elastic waves seem to be a viable means to detect delimitations, cracks and material degradation. Due to the complexity of such structures efficient numerical tools are called for. Several studies have shown that linear or quadratic pure displacement finite elements are not appropriate to resolve wave propagation problems. Both the mesh density and the spatial resolution needed to control the numerical dispersion are prohibitively large. Therefore, higher order finite element methods (p-FEM, SEM) are considered by the authors. One important goal is to simulate the propagation of guided ultrasonic waves in carbon/glass fiber reinforced plastics (CFRP, GFRP) or sandwich materials. These materials are typically deployed in aeronautical and aerospace application and feature a complex micro-structure. This micro-structure, however, needs to be resolved in order to capture effects like transmission, reflection and conversion of the different wave modes. It is known that using standard discretization techniques it is almost impossible to mesh the aforementioned heterogeneous materials without accepting enormous computational costs. Therefore, the authors propose to apply the finite cell method (FCM) and extend this approach by using Lagrange shape functions evaluated at a Gauss-Lobatto-Legendre grid. The latter scheme leads to the so called spectral cell method (SCM). Here, the meshing effort is shifted towards an adaptive integration technique used to determine the cell matrices and load vectors. Hence, a rectangular Cartesian grid can be used, even for the most complex structures. The functionality of the proposed approach will be demonstrated by studying the Lamb wave propagation in a two-dimensional plate with a circular hole. The perturbation is not symmetric with respect of the middle plane in order to introduce mode conversion. In the paper, an efficient method to simulate the elastic wave propagation in heterogeneous media utilizing the finite or spectral cell method is presented in detail.

An Explicit Model for Three-Dimensional Fluid-Structure Interaction using LBM and p-FEM

An explicit coupling model for the simulation of surface coupled fluid-structure interactions with large structural deflections is introduced. Specifically, the fluid modeled via the Lattice Boltzmann Method (LBM) is coupled to a high-order Finite Element discretization of the structure. The forces and velocities are discretely computed, exchanged and applied at the interface. The low compressibility of the Lattice Boltzmann Method allows for an explicit coupling algorithm. The proposed explicit coupling model turnes out to be accurate, very efficient and stable even for nearly incompressible flows. It was implemented in three software components: VirtualFluids (fluid), AdhoC (structure) and FSIsce (a communication library). The validity of the approach is demonstrated in two dimensions by means of comparing numerical results to measurements of an experiment. This experiment involves a flag-like structure submerged in the laminar flow field of an incompressible fluid where the structure exhibits large, geometrically non-linear, self excited, periodic motions. The methodology is then extended to three dimensions. Its performance is first demonstrated via the computation of a falling sphere in a pipe. The close correspondence of the results obtained by application of the numerical scheme compared to a semi-analytic solution is demonstrated. The proposed explicit coupling model is then extended to a plate in a cross flow. We verify the results by comparing them to results obtained by application of the commercial ALE-Finite Volume—h-FEM Fluid-Structure interaction solver Ansys Multiphysics. Additional examples demonstrate the applicability of the proposed methodology to problems of (arbitrarily) large deformations and of large scale.