scholarly journals Boundary integral equations for plane orthotropic bodies and exterior regions

2010 ◽  
Vol 8 (2) ◽  
Author(s):  
G. Szeidl ◽  
J. Dudra
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Orthotropic Material ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Exterior Regions ◽  
Direct Boundary Element Method ◽  
Element Method

Assuming linear displacements and constant strains and stresses at infinity, we re-formulate the equations of the direct boundary element method for plane problems of elasticity. We consider a body made of orthotropic material. The reformulated equations make it possible to attack plane problems on exterior regions without replacing the region by a bounded one.

Removing Non-Uniqueness in Symmetric Galerkin Boundary Element Method for Elastostatic Neumann Problems and its Application to Half-Space Problems

2020 ◽  
Vol 36 (6) ◽  
pp. 749-761
Author(s):  
Y. -Y. Ko
Keyword(s):  
Boundary Element Method ◽  
Rigid Body ◽  
Boundary Element ◽  
Integral Equations ◽  
Half Space ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Neumann Problems ◽  
Galerkin Boundary Element Method ◽  
Element Method

ABSTRACTWhen the Symmetric Galerkin boundary element method (SGBEM) based on full-space elastostatic fundamental solutions is used to solve Neumann problems, the displacement solution cannot be uniquely determined because of the inevitable rigid-body-motion terms involved. Several methods that have been used to remove the non-uniqueness, including additional point support, eigen decomposition, regularization of a singular system and modified boundary integral equations, were introduced to amend SGBEM, and were verified to eliminate the rigid body motions in the solutions of full-space exterior Neumann problems. Because half-space problems are common in geotechnical engineering practice and they are usually Neumann problems, typical half-space problems were also analyzed using the amended SGBEM with a truncated free surface mesh. However, various levels of errors showed for all the methods of removing non-uniqueness investigated. Among them, the modified boundary integral equations based on the Fredholm’s theory is relatively preferable for its accurate results inside and near the loaded area, especially where the deformation varies significantly.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

2017 ◽  
Vol 743 ◽  
pp. 158-161
Author(s):  
Andrey Petrov ◽  
Sergey Aizikovich ◽  
Leonid A. Igumnov
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Water Pressure ◽  
Boundary Integral ◽  
Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

Completed Double Layer Boundary Element Method

1994 ◽  
Author(s):  
Nhan Phan-Thien ◽  
Sangtae Kim
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Mesh Generation ◽  
Boundary Integral Equations ◽  
Direct Method ◽  
Boundary Integral ◽  
Deformation Field ◽  
Spatial Methods ◽  
Element Method

Despite the linearity of the Navier equations, solutions to complex boundaryvalue problems require substantial computing resources, especially in the so-called exterior problems, where the deformation field in the space between the inclusions to infinity must be calculated. In the traditional spatial methods, such as finite difference, finite element, or finite volume, this space must be discretized, perhaps with the help of "infinite" elements or a truncation scheme at a finite but large distance from the inclusions (Beer and Watson, Zienkiewicz and Morgan). There are two important limitations of spatial methods. The first is the mesh generation problem. To be numerically efficient, we must use unstructured mesh and concentrate our effort on where it is needed. Efficient two-dimensional, unstructured, automatic mesh generation schemes exist-one good example is Jin and Wiberg-but unstructured three-dimensional mesh generation is still an active area of research. The second limitation is much more severe: even a moderately complicated problem requires the use of supercomputers (e.g., Graham et al.). Since we are concerned with the large-scaled simulations of particulate composites, with the aim of furnishing constitutive information for modeling purposes, our system will possibly have tens of thousands of particles, and therefore the spatial methods are out of the question. We have seen how the deformation field can be represented by a boundary integral equations, either by a direct method, which deals directly with primitive variables (displacement and trciction) on the surface of the domain, or by the indirect method, where the unknowns are the fictitious densities on the surface of the domain. When the field point is allowed to reside on the surface of the domain, then a set of boundary integral equations results that relates only to the variables on the boundary (displacement and traction, or fictitious densities), and this is the basis of the boundary element method. The boundary is then discretized, and the integrals are evaluated by suitable quadratures; this then leads to a set of algebraic equations to be solved for the unknown surface variables.

scholarly journals Dynamic Stationary Response of Reinforced Plates by the Boundary Element Method

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
Luiz Carlos Facundo Sanches ◽  
Euclides Mesquita ◽  
Renato Pavanello ◽  
Leandro Palermo Jr.
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Mode Shapes ◽  
Plate Structures ◽  
Plane State ◽  
Reinforced Plate ◽  
Element Method

A direct version of the boundary element method (BEM) is developed to model the stationary dynamic response of reinforced plate structures, such as reinforced panels in buildings, automobiles, and airplanes. The dynamic stationary fundamental solutions of thin plates and plane stress state are used to transform the governing partial differential equations into boundary integral equations (BIEs). Two sets of uncoupled BIEs are formulated, respectively, for the in-plane state (membrane) and for the out-of-plane state (bending). These uncoupled systems are joined to form a macro-element, in which membrane and bending effects are present. The association of these macro-elements is able to simulate thin-walled structures, including reinforced plate structures. In the present formulation, the BIE is discretized by continuous and/or discontinuous linear elements. Four displacement integral equations are written for every boundary node. Modal data, that is, natural frequencies and the corresponding mode shapes of reinforced plates, are obtained from information contained in the frequency response functions (FRFs). A specific example is presented to illustrate the versatility of the proposed methodology. Different configurations of the reinforcements are used to simulate simply supported and clamped boundary conditions for the plate structures. The procedure is validated by comparison with results determined by the finite element method (FEM).

scholarly journals Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems

2017 ◽  
Vol 8 (1) ◽  
pp. 103-127
Author(s):  
A. Aimi ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Hyperbolic Partial Differential Equations ◽  
Weak Formulation ◽  
Exterior Problems ◽  
Element Method

Abstract Time-dependent problems modeled by hyperbolic partial differential equations can be reformulated in terms of boundary integral equations and solved via the boundary element method. In this context, the analysis of damping phenomena that occur in many physics and engineering problems is a novelty. Starting from a recently developed energetic space-time weak formulation for 1D damped wave propagation problems rewritten in terms of boundary integral equations, we develop here an extension of the so-called energetic boundary element method for the 2D case. Several numerical benchmarks, whose numerical results confirm accuracy and stability of the proposed technique, already proved for the numerical treatment of undamped wave propagation problems in several dimensions and for the 1D damped case, are illustrated and discussed.

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