Chemnitzer FEM-Symposium 1997

Vortragszusammenfassungen

Zur Lagrange-Interpolation einer Funktion benötigt man deren Stetigkeit. Für nicht-stetige Funktionen kann man den Scott-Zhang-Operator zur Interpolation verwenden. In seiner originalen Form ist er jedoch nur bedingt zur Interpolation auf anisotropen Finite-Elemente-Netzen geeignet. In diesem Vortrag sollen Modifikationen des Scott-Zhang-Operators vorgestellt werden, die anisotrope Abschätzungen der Ableitungen des zugehörigen Interpolationsfehlers gestatten.

Apel, Th.: Interpolation of non-smooth functions on anisotropic finite element meshes. Preprint SFB393/97-6, TU Chemnitz, 1997.

The typical limitations of standard wavelet techniques caused by complex domain geometries can be overcome when dealing with domains or manifolds that admit a natural partition into essentially disjoint parametric images of the unit d-cube. Based on a certain characterization of function spaces on the global domain or manifold in terms of a product space, wavelet bases are obtained by lifting well understood bases on the unit cube satisfying certain boundary conditions. It is indicated that this preserves all the desirable features of wavelet bases such as discrete norm equivalences and vanishing moments. As a consequence matrix compression and efficient preconditioning techniques are available. Moreover, the above mentioned characterization of function spaces provides a natural foundation for a domain decomposition technique which is well suited also for integral operators and allows one to exploit the full adaptive potential of wavelet expansions for elliptic problems of positive and nonnegative order.

Zur adaptiven Netzverfeinerung bei der Finiten Elemente Methode verwendet man häufig auf residualen Fehlerschätzern basierende Verfahren. Dabei treten Konstanten vor gewichteten Normen eines Volumenanteils und eines Sprungterms über Zwischenelmentkanten auf. Von diesen Konstanten weiß man, daß diese nur von der Form der Elemente abhängen und von der Ordnung sind, aber im allgemeinen nicht bekannt sind. Somit ist die Gewichtung beider Anteile im residualen Fehlerschätzer (meist durch den Faktor 1) rein heuristisch.

In dem Vortrag wollen wir zeigen, wie man mittels des Clement Interpolationsoperators Kenntnisse über diese beiden Konstanten gewinnen kann, d.h. wie man berechenbare Abschätzungen für diese Konstanten erhält. Desweiteren zeigen wir - durch die theoretischen Abschätzungen fundiert - voll numerische, berechenbare Abschätzungen für den Fehler in der Energienorm, die gegenüber den oben genannten um den Faktor 2-3 genauer sind. Unter anderem diskutieren wir die Parallelisierung dieser Fehlerschätzer und der daraus resultierenden adaptiven Netzverfeinerung.

Anhand numerischer Beispiele wird die Qualität der Fehlerschätzer gezeigt.

We present two artificially constructed hierarchical preconditioning methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on arbitrary unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner by decomposing functions on it are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of our methods which can be of enormous importance in the industrial engineering, when often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available. The approach can be transfered to the solution of real 3D-problems.

Die Identifikation von Materialparametern als abschließender und für die praktische Nutzbarkeit entscheidender Schritt der Materialmodellierung führt bei komplexem Werkstoffverhalten nicht zu akzeptablen Ergebnissen, wenn allein Experimente mit homogenen Spannungs- und Deformationszuständen ausgewertet werden. Für die akademische Forschung und die industrielle Praxis werden daher zunehmend Verfahren interessant, Materialparameter durch Analyse inhomogener mechanischer Felder zu identifizieren.

Es wird eine Methode zur Parameteridentifikation für elastisch-plastisches Materialverhalten mit isotroper und kinematischer Verfestigung und Anfangsanisotropie unter Auswertung experimentell ermittelter inhomogener Verschiebungsfelder von Vier-Punkt-Biege-Proben vorgestellt. Das inverse Problem wird durch nichtlineare Optimierung mit Hilfe von Gradientenverfahren numerisch gelöst. Innerhalb eines jeden Optimierungsschrittes erfolgt eine Finite-Element-Simulation des Experimentes zur Ermittlung der Vergleichswerte für die Verschiebungsfelder.

Ein wichtiger Bestandteil von Gradientenverfahren ist die Ermittlung der Ableitungen des Zustandsgrößenvektors nach den Optimierungsvariablen. Für das vorgestellte Problem bietet sich eine semianalytische Vorgehensweise zur Berechnung des Gradienten der Zielfunktion an, da der entsprechende Algorithmus eine zur FEM-Gleichgewichtsiteration analoge Struktur aufweist. Die effektive Materialparameteridentifikation auf der Basis der Analyse inhomogener mechanischer Felder erfordert große Rechenleistungen, wie sie z.B. durch massiv-parallele Rechner gegeben sind. Zudem wird deren Anwendung durch die natürliche Parallelisierbarkeit von großen Teilen des FEM-Algorithmus und der semianalytischen Sensitivitätsanalyse begünstigt.

Der Vortrag behandelt kurz die theoretischen Grundlagen des mechanischen Modells und der expliziten Verfahren zur Lösung von semidiskreten Bewegungsgleichungen. Danach wird das Konzept zur Parallelisierung des Verfahrens vorgestellt. Abschließend werden Beispiele diskutiert.

The Moving Finite Element method is an adaptive finite element technique for the solution of time-dependent partial differential equations in which the positions of the vertices of the triangulation are permitted to evolve as the solution develops. This talk will give a brief introduction to the method and will then describe a number of recent results in which optimal properties are proved for the locations of the vertices for certain classes of problem. The talk will finish with a description of some on-going research in which we are seeking to develop local, rather than global, strategies for the evolution of the mesh so as to reduce the computational expense of moving grid algorithms.

The numerical simulation of complex field problems by means of finite element discretizations leads, in general, to large scale systems of (non-) linear equations. The non-linearity can be treated by a linearization technique, such that the efficient solution of large scale systems of linear finite element equations is a basic problem. It is well-known that multi-grid and preconditioned conjugate gradient methods with multiplicative or additive multilevel preconditioners are the most efficient solvers on sequential and parallel computers. These solvers are optimal methods, i.e., the cost of arithmetical work for getting a solution with a relative accuracy is of the order (N - the number of unknowns).

In the talk, we describe the parallelization of these optimal methods applied to solve systems of finite element equations arising from the discretization of elliptic boundary value problems in two- and three-dimensional domains. The parallelization concept is based on a non-overlapping domain decomposition data distribution such that the algorithms can be implemented on MIMD parallel computers very well.

We show some advantages and disadvantages of the different parallel solvers and compare these solvers by numerical examples.

Some boundary value problems yield anisotropic solutions, e.g. solutions with boundary layers. If such problems are to be solved with the finite element method (FEM), anisotropically refined meshes can be advantageous. In order to construct these meshes or to control the error one aims at reliable error estimators.

For isotropic meshes such estimators are known but they fail when applied to anisotropic meshes. Estimators for rectangular (or cuboidal) anisotropic meshes were already investigated.

In this talk several error estimators are presented for tetrahedral or triangular meshes which offer a much greater geometrical flexibility. Some difficulties and conclusions will be discussed.

While orthogonal wavelets have primarily become known for their use in signal and image compression, in other application areas it is of advantage to have a compactly supported Riesz bases consisting e.g. of biorthogonal wavelets. Since it was established that wavelets as basis functions provide an asymptotic optimal preconditioning for linear systems stemming from a Galerkin method for elliptic p.d.e.s, there have been different approaches to exploit the potential of biorthogonal wavelets for additional requirements. Most of these are based on the idea to treat the boundary conditions separately. A wavelet-based approach also provides an asymptotic optimal preconditioning for the saddle point problems resulting from appending boundary conditions by Lagrange multipliers. In my talk I want to present newest theoretical and numerical results in this direction.

A nonoverlapping domain decomposition method with adaptive interface conditions [2] is considered for solving singularly perturbed second order elliptic boundary value problems with (stabilized) finite element methods. The approach allows for an a-priori mesh refinement in boundary (and interior) layer regions, in particular with anisotropic finite elements. We address in more detail the resolution of parabolic (or characteristic) layers which are of some importance in applications. The numerical analysis is based on both anisotropic interpolation estimates [1] and sharp estimates of derivatives [3]. Finally we draw some conclusions for the application of the proposed approach to the simulation of incompressible flow problems.

1

Apel, Th., Lube, G.: Anisotropic mesh refinement for singularly perturbed reaction diffusion problems, submitted to Appl. Numer. Math.

2

Auge, A., Lube, G., Otto, F.C.: A nonoverlapping domain decomposition method with adaptive interface conditions for elliptic problems, to appear in: Notes Numer. Meth. Fluids, Vieweg 1997

3

Roos, H.G.: Layer-adapted grids for singular perturbation problems, Preprint MATH-NM-03-1997, TU Dresden 1997

We analyze the h-p version of the boundary element Galerkin method for hypersingular integral equations of the first kind on open surfaces resulting from the Neumann problem for the Laplacian. We introduce countably normed spaces on the open surface containing all types of singularites arising from corners and edges. We show that all functions belonging to these countably normed spaces can be approximated exponentially fast in the -norm on geometric meshes using spaces of tensor products of piecewise polynomials (cf. Maischak 1995). Due to the tensor product structure we have highly anisotropic meshes. Using a regularity result (see Holm 1996) we show exponentially fast convergence of the Galerkin error in the -norm, i.e. energy-norm, for the h-p version on geometric meshes for piecewise analytic data.

Our numerical experiments confirm the theoretical results.

Die Verwendung moderner paralleler iterativer Gleichungslöser (CG mit hierarchischer Vorkonditionierung) ermöglicht es, insbesondere 3 dimensionale Deformationszustände mit hoher Effizienz numerisch zu simulieren. An Hand numerischer Experimente wird die Eignung dieser Techniken zur Lösung der bei der konsistenten Linearisierung geometrisch, (hyperelastischer) bzw. physikalisch (elastisch-plastischer) nichtlinearer Probleme entstehenden Gleichungssysteme diskutiert. Das zur Berechnung der Spannungen und deren Ableitungen im Fall plastischen Materialverhaltens verwendete Newton-Verfahren wird in seiner Verallgemeinerung für geometrische nichtlineare Probleme angegeben, sein quadratisches Konvergenzverhalten mittels obiger Beispiele illustriert und sein organischer Zusammenhang mit dem System der Materialgleichungen gezeigt.

This talk will be concerned with the anisotropic singular behaviour of the solution of elliptic boundary value problems near corners and edges.

First the anisotropic singular behaviour of the solution will be shortly described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions will be investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. The grading conditions will be described and illustrated. For the proof, new local interpolation error estimates in anisotropically weighted spaces are derived. Finally, a numerical experiment will be presented, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones. Here we shall compare ungraded meshes with three refined ones.

Apel, Th.; Nicaise, S.: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Preprint SFB393/97-7, TU Chemnitz, 1997. To appear in Math. Meth. Appl. Sci.

Für die Berechnung von grossen 3-dimensionalen Finite-Element-Problemen ist es vorteilhaft

• Adaptivität, um die Zahl der Freiheitsgrade möglichst klein zu halten
• parallele Algorithmen, um Zugriff auf die (potentiell unbegrenzten) Kapazitäten in Speicher und Rechenleistung von Parallelrechnern zu haben
• iterative Löser, da sie effizient parallelisierbar sind und Gleichungssysteme mit weniger Operationen (O(n*log(n)) als direkte Loeser (O(n*n)) lösen können.

Wir präsentieren einen adaptiven 3-D Verfeinerungsalgorithmus der reguläre, hierarchische Netze mit Quaderelementen ohne hängende Knoten erzeugt. Der Algorithmus ist in dem objekt-orientierten Programm PARAFEP implementiert. Die unstrukturierten Gleichungssysteme werden mit einem vorkonditioniertem Konjugierte Gradientenverfahren gelöst. Die Vorkonditionierung basiert auf Multilevel-Methoden. Beispiele sollen die Funktionalität und Effektivität belegen.

It is presented a robust multi-level-preconditioner for anisotropic elliptic differential equations of second order. The multi-level-preconditioner consists of semi-coarsening in one direction and line Gauss-Seidel relaxation in the opposite direction. The mesh size of the coarsest grid is the square root of the mesh size of the finest grid.

The corresponding additive Schwarz preconditioner is analyzed in detail. It is proved that the condition number of the preconditioner depends only on the meshsize and the smoothness of the coefficients of the equation in one direction. Furthermore, it is proved that the condition number is smaller than 178 (n+1), if the number n of levels is large enough or if the coefficients are constant in one direction. This property holds even if the equation is not -elliptic (e.g. ).

The multi-level-preconditioner is a robust preconditioner for linear equation systems which appear from anisotropic discretizations.

Die Kleinste-Quadrate-Methode führt als Minimierungsaufgabe auf ein elliptisches Problem, dessen FEM-Diskretisierung einfacher als die Behandlung der beim Galerkin-Verfahren entstehenden Sattelpunktaufgaben ist. Die entstehende Systemmatrix ist symmetrisch und positiv definit. Jedoch liegt hier die Schwierigkeit in der Konstruktion eines geeigneten Kleinste-Quadrate-Funktionals.

Die Gl.en 2.Ordnung für lineare (inkompressible) Elastizitätsprobleme (verallgemeinertes Stokes-Problem) können als Gl.en 1.Ordnung geschrieben werden. Dabei treten die Verschiebungen und deren Ableitungen (Spannungen) als Unbekannte auf. Die -Formulierung erfordert dann eine Glattheit der Lösung, die bei Vorhandensein von Ecken auf dem Gebietsrand i.a. nicht gegeben ist.

In einer numerischen Studie an einer -Ecke wird gezeigt, daß man zu guten Resultaten kommen kann, wenn man das Kleinste-Quadrate-Funktional in einem gewichteten -Raum nimmt, der die Lösung enthält.

In this lecture we shall discuss convection-dominated convection-diffusion problems of the type

equipped with corresponding boundary conditions such that as well exponential and parabolic boundary layers as interior layers can occur. We investigate with respect to the perturbation parameter uniformly stable finite element methods on a Shishkin mesh (in different norms).

Even if the mesh is correctly chosen the derivation of uniform error estimates requires new ingredients because Shishkin meshes do not satisfy the standard assumptions of the finite element technology, for instance. First, the element aspect ratio is large on some elements, and second, large variations of the element sizes are possible. But in some standard situations optimal uniform error estimates are available.

The definition of a correctly chosen mesh requires precise information on the behaviour of derivatives of the exact solution of the problem near layers. In many cases it is possible to derive the necessary information from asymptotic decompositions of the solution.

Numerical experiments prove that stable methods on Shishkin meshes work well also compared with standard adaptive approaches, for instance, which are not always able to detect layers. Unfortunately, robust error estimators with respect to a suitable norm so far are not available.

Multiscale bases like wavelets offer an versatile and efficient tool for adaptive approximation. In particular, they give rise to sparse discretization for integral equations and promise various applications in the numerical solution for partial differential equations. A central problem in this regard is the adaptation of multiscale bases to bounded domains and arbitrary manifolds or surfaces and the incorporation of boundary conditions.

We construct biorthogonal wavelet bases on domains and manifolds , which are given by the union of parametric images of the unit square (or cube). Following a domain decomposition strategy a basic step is the construction of biorthogonal wavelet bases on the unit square (resp. unit cube) satisfying boundary conditions on any given set of edges (resp. faces). The order of approximation d and the number of vanishing moments of the primal wavelets can be chosen. For primal spaces we choose tensor product spline spaces and the regularity of the dual functions is well controlled. A major tool for construction is the stable completion. Afterwards the functions are lifted to the corresponding parametric patches. Particular attention has to be paid for glueing the basis functions continuously together. In a first approach, biorthogonality is realized with respect to a modified inner product and the primal as well as the corresponding dual functions are globally continuous. This approach is not completely satisfactory for the multiscale discretization of operators of negative orders. In a second approach we go further realizing a complete decomposition of the manifold or domain by extension and restriction operators. It turns out that this approach is appropriate for a conform Galerkin discretization of second order partial differential operators as well as of integral operators negative order.

Eine naive FEM Diskretisierung von Elastizitätsproblemen auf dünnen Gebieten, sogenannten Platten oder Schalen, führt aufgrund von Locking-Effekten zu unbrauchbaren Ergebnissen. Mittels Strukturmechanik wird das 3D Problem auf ein System in der Ebene reduziert, z.B. auf die Plattengleichung nach Mindlin Reissner. Diese ist ein parameterabhängiges Problem, welches z.B. mittels reduzierter Integration robust diskretisiert wird.

Der erste Schwerpunkt des Vortrages ist die Konstruktion eines robusten Mehrgitterverfahrens für die Mindlin Reissner Gleichung. Die grundlegenden Komponenten sind Block-Glätter, geeignete Grobgitterdiskretisierungen und stetige Prolongationen. Der zweite Schwerpunkt ist die Übertragung dieser Technik auf dünne 3D Elemente. Diese können einfach für komplizierte Schalenstrukturen eingesetzt werden.

The character of convection dominanted, singularly perturbed boundary value problems requires their special numerical treatment in order to guarantee stability and to resolve existing layers with acceptable accuracy. In addition to discretization methods particularly developed for this aim, recently more and more attention is directed towards adapted triangulations of the computational domain. In this lection, an adaptive strategy based on an anisotropic refinement is presented for finite element methods. Starting from some a priori information about the location of layers, so-called hybrid meshes are constructed. By these meshes, the flexibility of unstructured meshes, good approximation properties in layers, and relatively simple rules for a posteriori anisotropic refinement are combined with each other. The efficiency of this procedure is demonstrated on selected numerical examples.

Gemischte Randwertprobleme können unter Benutzung von Steklov-Poincaré Operatoren als Variationsproblem zur Bestimmung einer der beiden Cauchy-Daten (Randverschiebungen oder Randspannungen) formuliert werden. Mittels Randintegralgleichungen können die Steklov-Poincaré Operatoren explizit ausgedrückt werden; eine direkte Auswertung der Galerkin-Gewichte ist aufgrund der Struktur der Operatoren jedoch nicht möglich.

Für die Galerkin-Diskretisierung der Variationsformulierung wird eine gemischte (hybride) Methode verwendet, die auf der Kollokationsdiskretisierung der Randintegralgleichung für ein Dirichlet-Problem beruht. Neben der nichtsymmetrischen Approximation des symmetrischen Steklov-Poincaré Operators werden symmetrische Methoden untersucht.

Für die Lösbarkeit des diskreten Systems wird eine Stabilitätsbedingung an die verwendeten Ansatzräume vorausgesetzt. Spektraläquivalenzungleichungen zur Konstruktion effizienter Vorkonditionierungen werden angegeben.

Die vorgestellten Diskretisierungsmethoden eignen sich sowohl für eine (herkömmliche) Parallelisierung als auch für den Einsatz in Gebietszerlegungsmethoden einschließlich der Kopplung mit Finiten Elementen.

A multi-structure is a three-dimensional body manifold represented as the union of sub-bodies with varying extensions in different directions. With other words we understand a multi-structure as a solid body which is an assembly of compact (3D) bodies, shells, plates, beams or other (three-dimensional) substructures.

In the first part of the talk we introduce a mathematical model in the framework of linear elasticity.

One of the main difficulties of mathematical modeling of such problems is the modeling of junctions between the different models. We circumvent this difficulty by using a primal hybrid formulation to model an elastic multi-structure. Our considerations are based on a rational description of models of structural mechanics. This allows a semi-discretization in the so called thickness direction by a structural mechanics model in every single sub-body combined with a finite element discretization of the structural mechanics model with respect to a lower dimensional reference manifold. Combined with the primal hybrid approach this allows a rational modeling to cover an extensive class of multi-structures.

The second part of the talk gives a discussion of the finite element solution of the model. We discuss

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finite element discretizations under the assumption of a valid LBB-condition,

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Hughes-Barbosa's circumventing LBB approach and

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a primal reformulation by elimination of the Lagrange (normal stress) parameter by Nitsche's method.

The BPX-like multilevel preconditioners are very efficient for second-order elliptic finite element discretizations, since they have a convergence rate which is independent of the discretization parameter, and the cost of arithmetical work per iteration step is proportional to the number of unknowns. P. Oswald has adapted these methods for discretizations of the fourth order biharmonic problem by different conforming and nonconforming elements.

In this talk we describe different shell models including Koiter's fourth order linear model and present some multilevel preconditioners for conforming and nonconforming finite element discretizations of these models.

We speak about the implementation of the preconditioners on MIMD parallel computers. Finally, we present some numerical results.

We present a parallel multigrid method for finite elements, realized with the software package UG. The basic software concepts are explained and the features of the finite element subsystem which supports a large number of discretizations are described.
This will be applied to elasto-plastic problems. Therefore, we introduce an abstract interface based on the Simo-Taylor linearization for rate-independent material laws: the time is discretized by the implicit Euler method and in each time step the material law is evaluated by a return mapping function. The consistent tangent moduli is determined by numerical derivation. We consider the Henkey model of perfect plasticity and a flow rule with isotropic hardening. The material history is stored in the Gauß points used for the numerical integration and can be evaluated independently. The displacement vector is stored in the interpolation points for the finite elements. They are used to solve the equilibrium equation in every Newton step. This is the only part of the algorithm with needs communication.

We consider in detail the parallel aspects of the multigrid method. Based on the parallel programming model DDD (Dynamic Distributed Data) by K. Birken an abstract formulation of parallel numerical algorithms is possible. Different consistency strategies for various smoothers are compared, leading to an efficient balance between the increasing communication and the loss of robustness. The performance of the algorithm is demonstrated by several examples.

We consider a posteriori error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods based on domain decomposition techniques. In particular, we restrict ourselves to nonconforming domain decomposition methods, and we use conforming P1 approximations on each subdomain. A so-called hierarchical error estimator obtained by the principle of defect correction in higher order finite element ansatz spaces and a localization by suitable hierarchical two-level splittings of these ansatz spaces is investigated as well as a residual based error estimator where the error is estimated by the dual norm of the residual. Additionally, a residual based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation. It turns out that the computational amount of the error estimator can be considerably reduced if we take the special structure of the Lagrange multiplier into account.