Boundary finite element method

INTERFACIAL STRESS CONCENTRATIONS NEAR FREE EDGES AND CRACKS BY THE BOUNDARY FINITE ELEMENT METHOD... 

In the present paper the Boundary Finite Element Method is presented as a boundary discretization method for the numerical investigation of interfacial stress concentrations in composite laminates. In contrast to the classical boundary element method, the element formulation is finite element based, which avoids the necessity of a fundamental solution. Comparative results from finite element calculations show good agreement both for the laminate free-edge effect and for the example of the stress concentrations near cracks in composite laminates. 

Keywords Boundary Finite Element Method, composite laminates, stress localization, laminate free-edge effect, free-edge stresses, crack problems, transverse matrix crack, numerical methods, boundary discretization... 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

The equation (33) and the subsequent equations show the main benefit of the Boundary Finite Element Method The field quantities inside the continuum can be calculated easily in closed-form analytical manner. 

In the following results are presented for the application of the Boundary Finite Element Method both for the case of the laminate free-edge effect and for the case of a single transverse matrix crack in the framework of linear elasticity theory. 

The Boundary Finite Element Method is presented as a numerical method which combines characteristics of the finite element method and the boundary element method. For certain geometric situations, this method allows an easy investigation of stress localization problems with less discretizational effort in comparison with the finite element method. For both the example of a transverse matrix crack and the example of the laminate fr( e-edg( effect, the results are shown to be in excellent agreement with comparative finite clement results. 

This paper compares experimental data for aluminium and steel specimens with two methods of solving the forward problem in the thin-skin regime. The first approach is a 3D Finite Element / Boundary Integral Element method (TRIFOU) developed by EDF/RD Division (France). The second approach is specialised for the treatment of surface cracks in the thin-skin regime developed by the University of Surrey (England). In the thin-skin regime, the electromagnetic skin-depth is small compared with the depth of the crack. Such conditions are common in tests on steels and sometimes on aluminium. 

D Finite Element / Boundary Integral Element method (TRIFOU)... 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

If the boundaiy is parallel to a coordinate axis any derivative is evaluated as in the section on boundary value problems, using either a onesided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line then more complicated expressions are needed and the finite element method may be the better method. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

Finite element methods [20,21] have replaced finite difference methods in many fields, especially in the area of partial differential equations. With the finite element approach, the continuum is divided into a number of finite elements that are assumed to be joined by a discrete number of points along their boundaries. A function is chosen to represent the variation of the quantity over each element in terms of the value of the quantity at the boundary points. Therefore a set of simultaneous equations can be obtained that will produce a large, banded matrix. 

Finite element methods are one of several approximate numerical techniques available for the solution of engineering boundary value problems. Analysis of materials processing operations lead to equations of this type, and finite element methods have a number of advantages in modeling such processes. This document is intended as an overview of this technique, to include examples relevant to polymer processing technology. 

The modeling of complex solids has greatly advanced since the advent, around 1960, of the finite element method [196], Here the material is divided into a number of subdomains, termed elements, with associated nodes. The elements are considered to consist of materials, the constitutive equations of which are well known, and, upon change of the system, the nodes suffer nodal displacements and concomitant generalized nodal forces. The method involves construction of a global stiffness matrix that comprises the contributions from all elements, the relevant boundary conditions and body and thermal forces a typical problem is then to compute the nodal displacements (i. e., the local strains) by solving the system K u = F, where K is the stiffness matrix, u the... 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

One nice feature of the finite element method is the use of natural boundary conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in Chang, M. W., and B. A. Finlayson, Inf. J. Num. Methods Eng. 15, 935-942 (1980), and Finla-son, B. A. (1992). The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

The trial functions in the finite element method are not limited to linear ones. Quadratic functions and even higher-order functions are frequently used. The same considerations hold as for boundary value problems The higher-order trial functions converge faster, but require more work. It is possible to refine both the mesh h and the power of polynomial in the trial function p in an hp method. Some problems have constraints on some of the variables. For flow problems, the pressure must usually be approximated by using a trial function that is one order lower than the polynomial used to approximate the velocity. 

Control volume method Finite element method Boundary element method and analytic element method Designed for conditions with fluxes across interfaces of small, well-mixed elements - primarily used in fluid transport Extrapolates parameters between nodes. Predominant in the analysis of solids, and sometimes used in groundwater flow. Functions with Laplace s equation, which describes highly viscous flow, such as in groundwater, and inviscid flow, which occurs far from boundaries. 

The equations are normally solved with a control volume finite element method. With these methods no boundary conditions are necessary at the outlets (i.e., it is implicitly assumed that there will always be an outlet at the point where the flow front merges and fills the last part of the mold). For a further discussion of boundary conditions and details about the numerical solution of the field equations (Eqs. 12.5 and 12.6) see [15,21,22-24]. 

All numerical techniques require application of sampling theory. Briefly stated, one chooses a representative sample of points within the region of interest and at each point attempts to calculate iteratively the most accurate solution possible, guided by self-consistency of local solutions with each other and with the specified boundary conditions. We describe two seemingly contrasting techniques finite-difference and finite-element methods (1,2). 

Stress calculations are carried out by the finite element method. Here, the commercial finite method code ABAQUS (Hibbit, Karlsson, and Sorensen, Inc.) is used. Other codes such as MARC, ANSYS are also available. To calculate the stresses precisely, appropriate meshes and elements have to be used. 2D and shell meshes are not enough to figure out stress states of SOFC cells precisely, and thus 3D meshes is suitable for the stress calculation. Since the division of a model into individual tetrahedral sometimes faces difficulties of visualization and could easily lead to errors in numbering, eight-comered brick elements are convenient for the use. The element type used for the stress simulation here is three-dimensional solid elements of an 8-node linear brick. In the coupled calculation between the thermo-fluid calculation and the stress calculation a same mesh model have to be used. Consequently same discrete 3D meshes used for the thermo-fluid analysis are employed for the stress calculation. Using ABAQUS, the deformations and stresses in a material under a load are calculated. Besides this treatment, the initial and final conditions of models can be set as the boundary conditions and the structural change can thus be treated. 

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