Effective numerical treatment of boundary integral equations: A formulation for three‐dimensional elastostatics

Lachat, Jacob; Watson, J. O.

doi:10.1002/nme.1620100503

Cited by 635 publications

(180 citation statements)

References 9 publications

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“…The integrals are defined, but they have to be suitably evaluated to avoid numerical problems. This topic has been studied intensively and an overview of the available techniques can be found in [78][79][80][81][82][83][84][85][86][87].…”

Section: Numerical Integrationmentioning

confidence: 99%

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A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

Maestre¹,

Pallarès²,

Cuesta³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. In this paper we describe a three-dimensional Isogeometric BEM in the time domain based on the T-spline and Non Uniform Rational B-Splines (NURBS) basis to study the free surface behavior. Traditionally, the Lagrange polynomials have been used to discretize the geometry and the BEMs variables. The Lagrange basis are discontinues across the elements although the physical variables are continuous. In dynamics problems with high mesh distortions this approach can produce numerical instabilities that can be quickly propagated. This problem could be overcome using T-spline or NURB basis. The main advantages of this approach are: (1) the control of the continuity and smoothness of the T-spline and NURBS basis, which makes the model numerically stable without the need of artificial smooth techniques;(2) the high order geometrical approximation by non-rational splines;(3) the refinement capabilities without affecting the geometry and BEM's variables; and (4) the direct integration with computer aid geometrical design tools. We use the concept of the Bézier extraction operation which provides an element point of view of the T-Spline and NURBS similar to the traditional finite element. The boundary integral Equation is solved at each time step by the GMRES algorithm and the time marching scheme is performed with a fourth-order Runge-Kutta method to update the model. Additionally, the hydrodynamic force is calculated by an auxiliary boundary equation. Some numerical benchmark examples are analysed to show the accuracy and the stability of the method 7967 J. Maestre, J. Pallarés and I. Cuesta

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Section: Numerical Integrationmentioning

confidence: 99%

“…The weak singularity is solved using a local change of variables over the element containing the collocation point, as proposed by Lachat and Watson [79]. Once regularized, these kernels can be numerically evaluated by applying the quadrature rule to each element in the parent domain.…”

Section: Numerical Integrationmentioning

confidence: 99%

A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

Maestre¹,

Pallarès²,

Cuesta³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…In this section we compare several numerical experiments with known analytical results for spheroids in uni form flows, using both standard integrations (based on the works by Lachat and Watson [28] and by Telles [45]) as well as the techniques exposed in Section 6.…”

Section: Numerical Verificationmentioning

confidence: 99%

“…2. A strategy should be chosen to integrate in regions which contain a singularity of type ð1=rÞ or ð1=r 2 Þ, such as the one presented in this paper, or one based on special quadrature techniques, such as those presented in [45] or [28], or any other technique to approximate Cauchy Principal value integrals (see, for example, [23]). …”

Section: Numerical Verificationmentioning

confidence: 99%

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Nonsingular isogeometric boundary element method for Stokes flows in 3D

Heltai

Arroyo

DeSimone

2014

Computer Methods in Applied Mechanics and Engineering

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While the finite element isogeometric analysis (FE IGA) has received most of the attention of the scientific community, only few results are available for boundary element methods (BEM) and boundary integral equations (BIE) in general. In 2D, there are some works on potential flows, such as [36,44], and on elastostatic analysis [40]. Three dimensional appli cations are of great interest in the maritime community, and there are some works on NURBS based panel methods to study, for example, marine propellers [25] or the wavemaking resistance problem [10]. Three dimensional IGA BEM linear elasticity has recently been studied in [20] while some work on shape optimization has been presented in [29]. Boundary element isogeometric analysis (IGA BEM) is very attractive for the solution of homogeneous elliptic PDEs, both in the interior and in the exterior cases, since it requires the solution of integral equations only on the boundary of the enclos ing (or enclosed) domain, which is typically the only information that is generated by standard CAGD tools. In contrast, FE IGA requires the extension of the computational domain inside (or outside) the enclosing CAGD surface, an outstand ing problem in IGA.While the dimensional reduction of the domain diminishes significantly both the number of degrees of freedom and the computational cost associated with the handling of the geometry, there are two main drawbacks which require special attention, and are common to all boundary integral discretizations: (i) the system matrices that are produced with a bound ary integral method are full, and (ii) the integrands contain ð1=rÞ singularities or worse (in three dimensions) around the source point, arising from the fundamental solutions in formulating the boundary integrals. Both issues have perhaps con tributed in keeping the IGA community away from boundary integral formulations.In the literature, the first issue is usually tackled by acceleration techniques such as the fast multipole method [39] (see [31] for a survey of this technique applied to BEM), which allow one to solve the system without assembling the full matri ces, and at the same time reduce the cost of the matrix vector multiplication from Oðn 2 Þ to OðnÞ (this technique has already been applied to Laplace problems in two dimensional IGA BEM in [44]). The second issue is more delicate and its solution depends highly on the degree of the discrete functional spaces and of the geometry description, remaining one of the most difficult aspects of the implementation of BEM. Different approaches are possible, (see, for example, [23]) but high order methods and curved geometries still remain challenging.Popular approaches to tackle singular integration involve a local change of variables for the quadrature rules, following the principle that the Jacobian of the transformation can be used to eliminate locally the kernel singularities. Examples of these methods are given by Lachat and Watson [28] and by Telles [45]. Both methods are elegant and produce accurate re sults, but t...

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Implementation of the Boundary Element Method in the Dynamics of Flexible Bodies

Amirouche

1996

Int. J. Numer. Meth. Engng.

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SUMMARYThis paper presents the implementation of the Boundary Element Method in the dynamics of flexible multibody systems. Kane's equations are used to formulate the governing boundary initial value problem for an arbitrary three-dimensional elastic body subjected to large overall base motion. Using continuum mechanics principles, direct boundary element incremental formulations are derived. The Galerkin approach was employed to generate the weighted residual statement which serves as a transitory point between continuum mechanics and boundary integral equations. By adapting the updated Langrangian formulation for large displacements analysis and using the Maxwell-Betti reciprocal theorem, integral representations for geometric stiffening were also derived. The non-linear terms were found to be functions of the time-variant stresses associated with the inertial forces at the reference configuration. The domain integrals arising from body forces (such as gravitational loads, inertia loads and thermal loads, etc.) are presented as DRM integrals (Dual-Reciprocity Method). Using the substructuring technique the elastic body is divided into several regions leading to a system of equations whose matrices are sparse (block-banded). The linearized equations of motion were discretized along the boundary of the body, and an algorithm for the integration involving the Houbolt method was used to establish an algebraic system of pseudo-static equilibrium equations. A Newton-Raphson-type iteration scheme was used to solve these discretized balance equations. To take advantage of the sparsity of the matrices, special routines were used to decompose and solve the resulting linear system of equations.An illustrative example is presented to demonstrate the validity of the method as well as how the effects of geometric stiffening effects are captured. The example consists of spin-up manoeuvre of a tapered beam attached to a moving base. The beam was modelled as two-dimensional plane strain problem divided into a number of substructures. Numerical simulation results show how the phenomenon of dynamic stiffening is captured by the present approach.

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Effective numerical treatment of boundary integral equations: A formulation for three‐dimensional elastostatics

Cited by 635 publications

References 9 publications

A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

Nonsingular isogeometric boundary element method for Stokes flows in 3D

Implementation of the Boundary Element Method in the Dynamics of Flexible Bodies

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