The heterogeneous multiscale finite element method FE-HMM for the homogenization of linear elastic solids

The objective of the present paper is a formulation of the Heterogeneous Multiscale Finite Element Method (FE-HMM) for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, for the first time, of a vector-valued field problem. The macro stiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Existing a-priori estimates are assessed and used for optimal micro-macro refinement strategies in the H¹-norm and the L²-norm. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

L 2 estimates for the finite element method for the Dirichlet problem with singular data

Summary In this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous Dirichlet condition on the boundary. In these problems the data are Borel measures. Using a quasiuniform mesh of finite elements and polynomials of degree 1, we prove that in two dimensions the convergence is of orderh inL ² and in three dimensions of orderh ^1/2.     

A finite element method dealing the singular points with a cut-off function

We consider an elliptic partial differential equation with input function inH ^-1. We assume that the type of singular part of solution is known and the solution is expressed as a sum of regular and singular term. We use a cut-off function to isolate the singular part and pose a variational problem to find the regular part of the solution. Then we have the solution by adding the regular solution and singular part. Error analysis and some example with computation will be given.     

A finite element method for cohesive crack modelling

The present contribution is concerned with the computational modelling of cohesive cracks, whereby the discontinuity is not limited to interelement boundaries, but is allowed to propagate freely through the elements. Inelastic material behaviour is described by a discrete constitutive law, formulated in terms of tractions and displacements at the surface. Details on the implementation and numerical examples are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive finite element method for singular parabolic equations

Summary. In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented. Mathematics Subject Classification (1991):65N22, 65F10, 65N30Supported by the Austrian Science Foundation (FWF) under the grant SFB F013}\and Walter Zulehner     

A finite element method for solving singular boundary-value problems

It is proved that under certain assumptions on the functions q(t) and f(t), there is one and only one function u₀(t) ∈ at which the functional

Dynamic crack propagation in a 2D elastic body: The out-of-plane case

Already in 1920 Griffith has formulated an energy balance criterion for quasistatic crack propagation in brittle elastic materials. Nowadays, a generalized energy balance law is used in mechanics [F. Erdogan, Crack propagation theories, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, pp. 498-586; L.B. Freund, Dynamic Fracture Mechanics, Cambridge Univ. Press, Cambridge, 1990; D. Gross, Bruchmechanik, Springer-Verlag, Berlin, 1996] in order to predict how a running crack will grow. We discuss this situation in a rigorous mathematical way for the out-of-plane state. This model is described by two coupled equations in the reference configuration: a two-dimensional scalar wave equation for the displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position derived from the energy balance law. We handle both equations separately, assuming at first that the crack position is known. Then the weak and strong solvability of the wave equation will be studied and the crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially. Using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an ordinary differential equation for the motion of the crack tip.     

A compact finite element method for elastic bodies

A nonconforming finite element method is described for treating linear equilibrium problems, and a convergence proof showing second order accuracy is given. The close relationship to a related compact finite difference scheme due to Phillips and Rose [1] is examined. A Condensation technique is shown to preserve the compactness property and suggests an approach to a certain type of homogenization.     

On the finite element method for elliptic problems with degenerate and singular coefficients

We consider Dirichlet boundary value problems for second order elliptic equations over polygonal domains. The coefficients of the equations under consideration degenerate at an inner point of the domain, or behave singularly in the neighborhood of that point. This behavior may cause singularities in the solution. The solvability of the problems is proved in weighted Sobolev spaces, and their approximation by finite elements is studied. This study includes regularity results, graded meshes, and inverse estimates. Applications of the theory to some problems appearing in quantum mechanics are given. Numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi-uniform meshes.

     

2D elastic analysis of the sandwich panel buckling problem: benchmark solutions and accurate finite element formulations

This paper is concerned with the elastic stability of a sandwich beam panel using classical elasticity. An exact solution for the buckling problem of a sandwich panel (wide beam) in uniaxial compression is presented. Various formulations that correspond to the use of different pairs of energetically conjugate stress and strain measures for the infinitesimal elastic stability of the sandwich panel are discussed. Results from the present two-dimensional analyses to predict the global and local buckling of a sandwich panel are compared with previous theoretical and experimental results. A new finite element formulation for the bifurcation buckling problem is also introduced. In this new formulation, terms that influence the buckling load, which have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. The formulation and results presented here can be used as a benchmark solution to establish the accuracy of numerical methods for computing the buckling behavior of thick, orthotropic solids.     

2D elastic analysis of the sandwich panel buckling problem: benchmark solutions and accurate finite element formulations

This paper is concerned with the elastic stability of a sandwich beam panel using classical elasticity. An exact solution for the buckling problem of a sandwich panel (wide beam) in uniaxial compression is presented. Various formulations that correspond to the use of different pairs of energetically conjugate stress and strain measures for the infinitesimal elastic stability of the sandwich panel are discussed. Results from the present two-dimensional analyses to predict the global and local buckling of a sandwich panel are compared with previous theoretical and experimental results. A new finite element formulation for the bifurcation buckling problem is also introduced. In this new formulation, terms that influence the buckling load, which have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. The formulation and results presented here can be used as a benchmark solution to establish the accuracy of numerical methods for computing the buckling behavior of thick, orthotropic solids.     

2D simulation of fluid-structure interaction using finite element method

This paper deals with pressure-based finite element analysis of fluid–structure systems considering the coupled fluid and structural dynamics. The present method uses two-dimensional fluid elements and structural line elements for the numerical simulation of the problem. The equations of motion of the fluid, considered inviscid and compressible, are expressed in terms of the pressure variable alone. The solution of the coupled system is accomplished by solving the two systems separately with the interaction effects at the fluid–solid interface enforced by an iterative scheme. Non-divergent pressure and displacement are obtained simultaneously through iterations. The Galerkin weighted residual method-based FE formulation and the iterative solution procedure are explained in detail followed by some numerical examples. Numerical results are compared with the existing solutions to validate the code for sloshing with fluid–structure coupling.     

Dynamic crack propagation in a 2D elastic body. The out-of plane case

We discuss the propagation of a running crack under shear waves in a rigorous mathematical way for a simplified model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u (y, t) and the one-dimensional crack tip trajectory h = h (t). We handle both equations separately, assuming at first that the crack position is known. Existence and uniqueness of strong solutions of the wave equation are studied and the crack-tip singularities are derived under the assumption that the crack is straight and moves tangentially. Using an energy balance law and the crack tip behaviour of the displacement fields we finally arrive at an ordinary differential equation for h (t), called equation of motion for the crack tip. We demonstrate the crack-tip motion with corresponding nonuniformly crack speed by numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling quasi-static crack growth with the embedded finite element method on multiple levels

The current work presents the multilevel approach of the embedded finite element method which is obtained by combining features of the method of domain decomposition with those of the standard embedded finite element method. The conventional requirement of fine mesh in a possible failure zone is rendered unnecessary with the new approach thereby reducing the computational expense. In addition, it is also possible to stop a propagating crack-tip in the middle of a finite element. In this approach, the finite elements at the failure-prone zone where cracks or shear bands, referred to as strong discontinuities which represent jumps in the displacement field, can form and propagate based on some failure criterion are treated as separate sub-boundary value problems which are adaptively discretized during the run time into a number of sub-elements and subjected to a kinematic constraint on their boundary. Each sub-element becomes equally capable of developing a strong discontinuity depending upon its state of stress. A linear displacement based constraint is applied initially which is modified accordingly as soon as a strong discontinuity propagates through the boundary of the main finite element. At the local equilibrium, the coupling between the quantities at two different levels of discretization is obtained by matching the virtual energies due to admissible variations of the main finite element and its constituent sub-elements. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A lumped mass finite element method for vibration analysis of elastic plate-plate structures

The fully discrete lumped mass finite element method is proposed for vibration analysis of elastic plate-plate structures. In the space directions, the longitudinal displacements on plates are discretized by conforming linear elements, and the transverse displacements are discretized by the Morley element. By means of the second order central difference for discretizing the time derivative and the technique of lumped masses, a fully discrete lumped mass finite element method is obtained, and two approaches to choosing the initial functions are also introduced. The error analysis for the method in the energy norm is established, and some numerical examples are included to validate the theoretical analysis.     

Hexahedral finite element with an open crack

A method of creating the stiffness matrix of a hexahedral eight-node finite element with a single, nonpropagating, transverse, one-edge crack at half of its length is presented in this paper. The crack was modelled by adding an additional flexibility matrix to that of the noncracked element. The terms of the additional matrix have been calculated by use of the laws of fracture mechanics. Employing the elaborated element a numerical test has been worked out, the results of which are compared with the data of analytical solutions accessible in the literature, and a high conformity with them has been obtained. The element presented in the paper may be applied to the static and dynamic analysis of different types of structural elements with material defects in the form of cracks. The described method of creating the stiffness matrix of the element allows to create different kinds of finite elements with cracks provided that the stress intensity factors for a given type of crack are known.     

Evaluation of singular integrals for anisotropic elastic boundary element analysis

To improve the numerical evaluation of weakly singular integrals appearing in the boundary element method, a logarithmic Gaussian quadrature formula is usually suggested in the literature. In this formula the singular function is expressed in terms of the distance between source point and field point, which is a real variable. When an anisotropic elastic solid is considered, most of the existing fundamental solutions are written in terms of complex variables. When the problems with holes, cracks, inclusions, or interfaces are considered, to suit for the shape of the boundaries usually a mapping function is introduced and then the solutions are expressed in terms of mapped complex variables. To deal with the trouble induced by the complex variables, in this study through proper change of variables we develop a simple way to improve the evaluation of weakly singular integrals, especially for the problems of anisotropic elastic solids containing holes, cracks, inclusions, or interfaces. By simple matrix expansion, the proposed method is extended to the problems with piezoelectric or magneto-electro-elastic solids. By using the dual reciprocity method, the proposed method employed for the elastostatic fundamental solution can also be applied to the elastodynamic analysis.