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## Über dieses Buch

The first international symposium on mathematical foundations of the finite element method was held at the University of Maryland in 1973. During the last three decades there has been great progress in the theory and practice of solving partial differential equations, and research has extended in various directions. Full-scale nonlinear problems have come within the range of nu­ merical simulation. The importance of mathematical modeling and analysis in science and engineering is steadily increasing. In addition, new possibili­ ties of analysing the reliability of computations have appeared. Many other developments have occurred: these are only the most noteworthy. This book is the record of the proceedings of the International Sympo­ sium on Mathematical Modeling and Numerical Simulation in Continuum Mechanics, held in Yamaguchi, Japan from 29 September to 3 October 2000. The topics covered by the symposium ranged from solids to fluids, and in­ cluded both mathematical and computational analysis of phenomena and algorithms. Twenty-one invited talks were delivered at the symposium. This volume includes almost all of them, and expresses aspects of the progress mentioned above. All the papers were individually refereed. We hope that this volume will be a stepping-stone for further developments in this field.

## Inhaltsverzeichnis

### Nonlinear Shell Models of Koiter’s Type

Abstract
We describe, and we discuss the merits of, a two-dimensional nonlinear shell model analogous to a model proposed by W.T. Koiter in 1966, where the exact change of curvature tensor is suitably modified. A first interest of this model, from the computational viewpoint, is that the resulting stored energy function becomes a polynomial with respect to the unknown components of the deformation field and their partial derivatives.
A second interest of this model is its amenability to a formal asymptotic analysis of its solution, with the thickness as the “small” parameter. Such an analysis yields exactly the same conclusions as the formal asymptotic analysis of the solution of the three-dimensional equations, thus providing a justification of the proposed model.
Philippe G. Ciarlet

### A Survey of Stabilized Plate Elements

Abstract
We present two families of finite element methods for the Reissner- Mindlin plate model. The families are based on a stabilized formulation which circumvents the requirement that the finite element spaces should satisfy the Babuška- Brezzi conditions. In the first family the polynomial order of the basis functions for the deflection is one higher than that for the rotation. In the second family the stabilization is combined with the MITC interpolation technique, which enables equal order basis functions. We review the stability and error estimates which show that the methods are ”locking-free” and optimally convergent.
Mikko Lyly, Rolf Stenberg

### Prediction of the Fatigue Crack Growth Life in Microelectronics Solder Joints

Abstract
In order to predict the crack growth life in microelectronics solder joints, an FE A (finite element analysis) program employing a new scheme for crack growth analysis is developed. Also some experimental data necessary for the practical application of this program are obtained. Above all, the data related to the crack growth rate play a key role and are obtained in terms of the maximum opening stress range Δσθ max as
$${{da} \mathord{\left/ {\vphantom {{da} {dN = }}} \right. \kern-\nulldelimiterspace} {dN = }}\beta \left[ {\Delta \sigma _{\theta max} - \gamma } \right]^\alpha ,$$
where α = 2.0 and β = 2.5 x 10-9 mm 5/N 2 are independent of the test conditions, and γ is dependent on the solder material. The calculated values of the crack growth life by the FEA are in good agreement with the experimental ones. This indicates at the same time that the crack growth rate and path are certainly controlled, through the above equation, by Δσθ max measured at a certain radial distance from the crack tip.
Ken Kaminishi

### Multi-phase Flow with Reaction

Abstract
In order to study the effects of spilled oil on coastal ecosystem, multiphase flow with reaction is modeled mathematically. In this procedure, Discontinuous Interface Generating Method plays an important role to formulate the decomposition phenomena of oil into water and soluble components. This mathematical model is numerically solved by use of finite difference method and the numerical results are presented.

### Universal and Simultaneous Solution of Solid, Liquid and Gas in Cartesian-Grid-Based CIP Method

Abstract
We present a review of the CIP method that is known as a general numerical solver for solid, liquid and gas. This method is a kind of semi-Lagrangean scheme and has been extended to treat incompressible flow in the framework of compressible fluid. Since it uses primitive Euler representation, it suits for multi-phase analysis. The recent version of this method guarantees the exact mass conservation even in the framework of semi-Lagrangean scheme. Comprehensive review is given for the strategy of the CIP method that has a compact support and subcell resolution including front capturing algorithm with functional transformation. Some practical applications are also reviewed such as milk crown or coronet.
Takashi Yabe

### Subgrid Phenomena and Numerical Schemes

Abstract
In recent times, several attempts have been made to recover some information from the subgrid scales and transfer them to the computational scales. Many stabilizing techniques can also be considered as part of this effort. We discuss here a framework in which some of these attempts can be set and analyzed.
Franco Brezzi, Donatella Marini

### Two Scale FEM for Homogenization Problems

Abstract
We analyze generalized Finite Element Methods for the numerical solution of elliptic problems with coefficients or geometries which are oscillating at a small length scale ɛ. Two-scale elliptic regularity results which are uniform in ɛ are presented. Two-scale FE spaces are introduced with error estimates that are uniform in ɛ. They resolve the ɛ scale of the solution with work independent of ɛ and without analytical homogenizations. Numerical experiments confirming the theory are presented.
Christoph Schwab

### Numerical Analysis of Electromagnetic Problems

Abstract
We give theoretical and computational overview of numerical analysis of the finite element methods for electromagnetics. In particular, theoretical comments on the edge and face elements, frequently employed in the finite element discretizations, are given. Moreover, we present some iteration methods which are effective to solve discrete equations arising from finite element methods.
Fumio Kikuchi

### A-priori Domain Decomposition of PDE Systems and Applications

Abstract
Domain Decomposition has been extensively studied as a tool for parallel computing. But in many cases the problem posed includes domain decomposition in its statement. For these the necessary numerical analysis is different because domain decomposition is not only at the discrete level but also on the continuous problem. Therefore non-matching grids for their numerical solutions is more natural, but requires new error estimates.
Our main purpose is to compute with the data of Virtual Reality. In this paper we shall review earlier works, including our own[9][10][11] and we shall present the project freefem3d.
S. Delpino, J. L. Lions, O. Pironneau

### A One Dimensional Model for Blood Flow: Application to Vascular Prosthesis

Abstract
We investigate a one dimensional model of blood flow in human arteries. In particular we consider the case when an abrupt variation of the mechanical characteristic of an artery is caused by the presence of a vascular prosthesis (e.g. a stent). The derivation of the model and the numerical scheme adopted for its solution are detailed. Numerical experiments show the effectiveness of the model for the problem at hand.
Luca Formaggia, Fabio Nobile, Alfio Quarteroni

### Essential Spectrum and Mixed Type Finite Element Method

Abstract
In the error analysis of finite element method, the inf-sup condition or the uniform lifting property plays an important role. In this paper, we discuss the relationship between the uniform inf-sup condition and the essential spectrum of the operator that appears in the problem. In general, one can not expect the convergence of the finite element approximation due to the spectral pollution that stems from the inappropriate mixing of the eigen-subspaces that correspond to two distinct components of the essential spectrum. As examples of our consideration, we treat the Stokes problem, mixed approximations of elliptic problems and a structural-acoustic coupling problem. In these problems, two distinct components might appear in the essential spectrum of the corresponding operators.
Takashi Kako, Haniffa M. Nasir

### Can We Trust the Computational Analysis of Engineering Problems?

Abstract
Computational science in general and computational mechanics in particular addresses physical and engineering reality with respect to some particular goals. These goals must be clearly specified. They are usually to get good qualitative or quantitative information about reality. The admissible quality of required information should be characterized.
I. Babuška, T. Strouboulis

### Numerical Computations for Ill-conditioned Problems by Multiple-Precision Systems

Abstract
We propose a use of some multiple-precision systems for numerical analysis of ill-conditioned problems, and we show efficiency of the systems through numerical examples. We also introduce the F-system which is a fast multiple-precision system designed by one of the authors.
Yuusuke Iso, Hiroshi Fujiwara, Kimihiro Saito

### Numerical Verification Methods for Solutions of Free Boundary Problems

Abstract
In this paper, we consider numerical techniques which enable us to verify the existence of solutions for the free boundary problems governed by two kinds of elliptic variational inequalities(EVIs). Based upon the finite element approximations and the explicit a priori error estimates for some simple EVIs, we present effective verification procedures that, through numerical computation, generate a set which includes exact solutions. We describe a survey of the previous works as well as show some newly obtained results up to now.
Mitsuhiro T. Nakao, Cheon Seoung Ryoo

### Pattern Formation of Heat Convection Problems

Abstract
We consider the Rayleigh-Bénard problem of the heat convection in the horizontal strip with the stress free boundary condition for the velocity and Dirichlet boundary condition for the temperature. We examine the pattern formation of the roll type solution, the rectangular type solution and the hexagonal type solution and see the stability of them and a better bifurcation diagram for the full system by using numerical computations.
Takaaki Nishida, Tsutomu Ikeda, Hideaki Yoshihara

### Mathematical Modeling and Numerical Simulation of Earth’s Mantle Convection

Abstract
Rheology and geometry are two important factors in the Earth’s mantle convection phenomenon. That is, the viscosity is strongly dependent on the temperature and the phenomenon occurs in a spherical shell domain. Focusing our attention on these two factors, we describe a total approach of numerical simulation of the Earth’s mantle convection, i.e., mathematical modeling, mathematical analysis, computational scheme, error analysis, and numerical result.
Masahisa Tabata, Atsushi Suzuki

### Theoretical and Numerical Analysis on 3-Dimensional Brittle Fracture

Abstract
In this paper, we propose a mathematical formulation of 3-dimensional quasi-static brittle fracture under varying loads. We give a precise formulation of internal cracking and surface cracking in 3D elastic bodies. In Sections 2 and 3, we provide geometrical results and results on Sobolev spaces. Most criteria in fracture mechanics are based on Griffith’s energy balance theory, which is explained briefly in Section 5. G J-integral is proposed by the author (1981), which is a generalization of J-integral widely used in 2D fracture problems. G J-integral expresses the variation of energies with respect to crack extensions and relates to Griffith’s energy balance theory. Under varying loads, we cannot use Griffith’s energy balance theory directly, however G J-integral is applicable to such cases too. For practical use, we must study the combination with numerical calculation. In the last section, we give an error estimate for finite element approximation of G J-integral.
Kohji Ohtsuka

### Exploiting Partial or Complete Geometrical Symmetry in Boundary Integral Equation Formulations of Elastodynamic Problems

Abstract
Procedures based on group representation theory, allowing the exploitation of geometrical symmetry in symmetric Galerkin BEM formulations of 3D elastodynamic problems, are developed. They are applicable for both commutative and noncommutative finite symmetry groups and to partial geometrical symmetry, where the boundary has two disconnected components, one of which is symmetric.
Marc Bonnet

### A New Fast Multipole Boundary Integral Equation Method in Elastostatic Crack Problems in 3D

Abstract
This paper discusses a formulation and its applications of the new Fast Multipole Method (FMM) to three-dimensional Boundary Integral Equation Method (BIEM) in elastostatic crack problems. It is shown, through numerical experiments, that the new FMM is more efficient than the original FMM.
Ken-ichi Yoshida, Naoshi Nishimura, Shoichi Kobayashi

### Computational Crack Path Prediction and the Singularities in Elastic-Plastic Stress Fields

Abstract
The paper justifies the assumption that the exponent of the first term in asymptotic expansion of two-dimensional stresses at a crack tip of elastic-plastic body is independent of the angle θ in polar coordinates. First we discuss the case of a total deformation theory and then apply the idea used there to an incremental theory. These results can be effectively used to show the validity of a procedure used in computational crack path prediction for elasic-plastic bodies. In Appendix we show that, if the ” ĵ-integral” does not vanish, the exponent is independent of the load parameter t too, and equal to -½ for stational cracks in the material with hardening, as is seen in elastic stresses.
Tetsuhiko Miyoshi

### Backmatter

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