Boundary element based formulations for crack shape sensitivity analysis

Controlling errors in the process of automating boundary element method analysis

Direct differentiation formulation for boundary element shape sensitivity analysis of axisymmetric elastic solids

A new boundary element method (BEM) based method is described for the design of coils for magnetic resonance imaging (MRI) systems. BEM is an effective approach for solving the electromagnetic forward problem and has been used in the design of MRI gradient coils. However, BEM-based gradient coil design faces an ill-posed mathematical problem, which is conventionally handled by means of a Lagrange multiplication method. This work attempts to improve the BEM method for MRI coil designs by applying the Tikhonov regularization scheme to solve the ill-posed matrix system formulated by the BEM forward model. The objective of the study is not to design some specific gradient or radio-frequency (RF) coils for MRI system, but to discuss the design scheme with practical regularization and constraints. The proposed approach was explained in the design of MRI coils including biplanar transverse gradient coils and RF phased array coils. With the consideration of the practical engineering requirements, physical constraints such as wire intervals are transformed into mathematical constraints and formulated into BEM equations. The examples demonstrate that the proposed method is efficient and flexible for the design of MRI coils with arbitrary geometries and engineering constraints.

This paper proposes a numerical model of cooling analysis of an injection mold that has an embedded insert. The Boundary Element Method (BEM) is employed because of its popular usage in this area. The key problem in model building is how to involve the independent surface between mold and insert into the BEM equations and how to give the natural boundary conditions on it. This problem is solved by using coupling BEM which provides an attractive alternative in that the natural boundary condition is shifted to the surface between part and mold. So, instead of using the finite differentiation method, the boundary condition can be determined analytically. This is useful in improving the accuracy of cooling analysis.

This paper presents an identification method of crack shapes on eddy-current testing. In order to use the impulse excitation, numerical analysis is performed by the boundary element method with Laplace transform. The shapes of cracks are identified directly from the solutions of the integral equation of boundary element method. The proposal method by the singular value decomposition and fuzzy regularization method are applied into the shape identifications of various crack widths and heights to examine our proposed method.

Several new formulations for structural design sensitivity analysis (DSA) of elastic solids are presented that consistently account for steady and transient thermal loading, along with centrifugal and gravity loading. The approach employs implicit differentiation of the discretized boundary integral equations and facilitates the production of accurate and efficient computational procedures that obviate the need for matrix factorizations in the DSA step. For centrifugal, gravity and general thermal loading, a new DSA concept employing a particular integral technique is developed. For centrifugal, gravity and steady thermal effects, a surface integration approach is also presented, and the two methods compared. Alternative strategies for modelling the behaviour of temperature distributions of evolving shapes undergoing shape optimization and the ramifications of these strategies in the DSA process are also discussed. It is shown that, for certain classes of thermoelastic problems, shape optimization can be performed without the need to incorporate thermal analysis or DSA into the overall process. A thermoelastic surface stress sensitivity analysis formulation is presented that does not require additional numerical integration. A new thermoelastic sharp corner traction component consistent sensitivity analysis formulation is also presented. This technique is shown to work for any initial stress distribution. Numerical sensitivity results are presented for many problems with analytical solutions.

Analysis of crack problems in functionally graded materials under thermomechanical loading using graded finite elements

An incremental finite element formulation for structural sensitivity analysis or nonlinear (possibly path dependent) systems with fixed overall shape is proposed. Both the direct differentiation and adjoint variable methods, as well as a mixed differentiation-adjoint technique are employed to evaluate 1st- and 2nd-order sensitivity increments during the load step. Numerical illustration is given.

A plasticity model with yield limit degradation is implemented in a boundary element program to study the fracture behavior of quasi-brittle materials. A special integration method is applied to deal with the singular integrations encountered in the volume integrals over internal cells. Strain-softening damage localizations are investigated. It is found that the damage tends to localize into a zone of one cell wide which leads to an incorrect result when different cell meshes are used in the analysis. Some sort of localization limiter has to be incorporated into the analysis in order to achieve meaningful results. INTRODUCTION In the simulation of fracture behavior of quasi-brittle materials such as concrete or ceramics, it is essential to take into account the evolution of a relative large fracture process zone forms in front of the crack tip. The process zone consists of hundreds of microcracks and its overall behavior can generally be simulated either by discrete-cracking models such as the fictitious crack model [4, 5] or smeared-cracking models such as strain-softening damage models [2, 3]. In this paper the fracture process zone is simulated by a plasticity model with yield limit degradation. When the stresses near the crack tip reach the yield surface, the yield surface will contract rather than expand. This results in a decrease in stresses during an increase in strains, i.e., a strain softening. The decrease in stresses simulates a material damage caused by mainly the formation of microcracks. Difficulty has been encountered in the finite element analysis involving strain-softening behavior. As different meshes are used in an analysis, Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 304 Boundary Elements different results are obtained. The results are not objective with respect to finite element meshes. Also, the energy dissipation of a structure at failure decreases with the refinement of the finite element mesh. It implies that the structure will fail at near zero energy dissipation when a very fine mesh is used. This is in contradiction with the reality. All these problems are considered due to damage localization [1,2,3]. In this paper, the plasticity model with yield limit degradation is incorporated into a boundary element program to study the damage localization problems. Many publications on the boundary element analysis of elastoplastic problems [7-14] have been seen since the first paper on the problem published in 1971 by Swedlow and Cruse [6]. However, the literature on the study of damage localization problems using boundary element approach is rarely seen. The initial stress boundary element method is applied to deal with the nonlinear damage problems, For localization problems, damage tends to localize into a small region of the body, and thus the internal cells required for integration purposes are needed only in a small region rather than the entire body. A special semi-analytical integration method for triangular cell is used to carry out the volume integration over the cell that contains a singular node. A rectangular panel with three different cell meshes are analyzed. It is shown that the results are not objective to the cell meshes even though all other conditions are kept the same. BOUNDARY ELEMENT FORMULATION FOR ELASTOPLASTIC ANALYSIS For elastoplastic problems based on small strain theory, the basic equations in incremental form are as follows: The governing equations are da^ + db* = 0 (1) with boundary conditions du;(x) = (x) x e I (2) cR (x) x e I I + T^T and total strain increment de^ = de^ 4de^= (du + dUjJ/2 (3) where de^ and de^ are elastic and plastic strain increments. T is the boundary of body 0 under consideration, da^ ,db; ,du;, dr; are the stress, body force (per unit volume), displacement, and traction increment respectively. du ± and df j are prescribed displacements and tractions on boundary I and r%. Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 305 The yield criterion is F(a%,k)= f(a$ <Kk) = 0 (4) where F and f are yield functions, $ is a work hardening function, and k is the work hardening parameter. The constitutive equations are de'u) = d<^ do, = DV^U (5) where D^= &pvl(l-2v)] 6fo + M<Wji + W

Stress Analysis using Boundary Elements

In this article, an application of a current sheet as a finite element in the boundary element method (BEM) is described. A detailed procedure for mathematical and software implementation of a current sheet element (CSE) as a finite boundary element (FBE) in BEM analysis is presented. For a simple generic magnetic circuit with a stator and a rotor, the calculated magnetic field and torque values are compared with the values obtained by two different finite-element method (FEM)-based simulation programs showing good agreement; however, elucidating the fact that even by employing professional FEM tools, on fairly simple problems, accurate determination of forces and torques is not necessarily guaranteed. Certain aspects of using the Maxwell stress tensor (MST) for the determination of forces and torques on the rotor domain are discussed. Since the primary magnetic field sources are currents in a set of conductors of the circular cross section, analytical expressions are used for the calculation of the primary field. Due to the selected geometry of the FBEs, i.e., CSEs, after a numerical determination of their currents by BEM, analytical expressions are employed for the calculation of FBE fields as well.

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

&lt;div class="htmlview paragraph"&gt;Radiated noise from various components is of increasing concern for today's automotive manufacturers. Traditionally, this analysis was usually performed with Boundary Element Method (BEM) software. Unfortunately, there is no sensitivity or optimization capability within any known BEM code. This paper addresses the problem by describing the software developed by CDH, i.e., OPT and VAO, which combine the structural FEM analysis &lt;i&gt;and&lt;/i&gt; optimization with BEM analysis such that one can perform analysis and optimization very quickly and efficiently. A sample air cleaner problem is described which illustrates the minimization of radiated noise at a predefined far-field positions.&lt;/div&gt;

An efficient numerical scheme for obtaining the two dimensional impurity profile and structural features of the semiconductor devices is proposed. The finite element method (FEM) is used to solve the drift-diffusion equation of impurities in silicon. The thermal oxidation using a model based on steady-state oxidant diffusion is dealt with the boundary element method (BEM). Both methods are employed for calculating the complicated geometry and the boundary conditions in the simulation, where these methods are simultaneously used for the oxidizing condition. For the finite element analysis, mass lumping is introduced to get the stable result. The matrix obtained is asymmetric and irregular sparse, the matrix is solved by the Conjugate Gradient-Squared method (CGS) with preconditioning-conditioning (Incomplete LDU factorization). Thus a more economical calculation method is obtained than the conventional FEM and BEM analysis, using Gaussian elimination method.

A direct traction BIE approach for three-dimensional crack problems