WIT transactions on modelling and simulation, Jun 11, 2013
The double-layer potential matrix H of the conventional, collocation boundary element method (CBE... more The double-layer potential matrix H of the conventional, collocation boundary element method (CBEM) is singular, as referred to a static problem in a bounded continuum. This means that no rigid body displacements can be transformed between two different reference systems-and that unbalanced forces are to be excluded from a consistent linear algebra contragradient transformation. The properties of H T may become quite informative, as they reflect the topology (concavities, notches, cracks, holes) of the discretized domain as well as material non-homogeneities, which comes from the fact that local stress gradients can be represented by fundamental solutions only in a global sense. Symmetries and antisymmetries are also evidenced in N (H T) as well as the spectral properties related to simple polynomial solutions that one may propose as patch tests. In the usual implementations of the CBEM with real fundamental solutions, all eigenvalues λ of H are real, λ ∈ R, 0 ≥ λ < 1 for a bounded domain. This means that H is a contraction-a paramount mechanical feature that comes up naturally in the frame of a virtual work investigation of Kelvin's (singular) fundamental solution and is resorted to in a simplified variational implementation of the boundary element method.
WIT transactions on engineering sciences, Aug 2, 2022
This paper is part of a research work to implement, test, and apply a novel numerical tool that c... more This paper is part of a research work to implement, test, and apply a novel numerical tool that can simulate on a personal computer and in just a few minutes a problem of potential or elasticity with up to tens of millions of degrees of freedom. The first author's group has already developed their own version of the fast multipole method (FMM) for two-dimensional problems, which relies on a consistent construction of the single-layer potential matrix of the collocation boundary element method so that ultimately only polynomial terms (as for the double-layer potential matrix) are required to be integrated along generally curved segments related to a given field expansion pole. The core of the present paper is the mathematical assessment of the double expansions needed in the 3D FMM. The 3D implementation is combined with a particular formulation for linear triangle elements in which all integrations for adjacent source point and boundary element are carried out analytically. As a result, numerical approximations are due exclusively to the FMM series truncations. This allows isolating and testing truncation errors incurred in the series expansions and thus for the first time properly assessing the mathematical features of the FMM, as illustrated by means of two examples. Adaptive numerical quadratures as well as the complete solution of a mixed boundary problem using a GMRES solver, for instance, are just additional tasks and, although already implemented, are not reported herein.
A boundary integral implementation generally involves the evaluation of singular and quasi-singul... more A boundary integral implementation generally involves the evaluation of singular and quasi-singular integrals. Dealing with them requires elaborated codes, which quite frequently results in prohibitive computational effort as the price of adequate numerical accuracy. In the frame of a research line in progress at PUC-Rio, a simple and accurate scheme has been developed for the treatment of general singular linear integrals. It is now being implemented for double integrals. This technique makes use of fixed abscissas, as taken from Gauss-Legendre quadrature, with different weight sets calculated for different singularities — a procedure that demands low computational effort. This feature allows the development of robust codes, in which integration demands low computational effort and yields high precision. The core of the present contribution is the introduction of an adequate scheme for the global-to-local transformation of the coordinates of a given singularity pole. This transformation may be used in connection with polar coordinates, as singular integrals are usually dealt with. In the paper, however, it is an essential feature of the unified integration procedure the authors are proposing. The technique is straightforward to implement, independently of the degree of singularity or quasi-singularity.
Lecture notes in computational science and engineering, 2017
We present a simple, PDE-based proof of the result [17] by M. Johnson that the error estimates of... more We present a simple, PDE-based proof of the result [17] by M. Johnson that the error estimates of J. Duchon [11] for thin plate spline interpolation can be improved by h 1/2. We illustrate that H-matrix techniques can successfully be employed to solve very large thin plate spline interpolation problems.
WIT transactions on modelling and simulation, Sep 8, 2014
This is the sequel of a paper presented at the BEM/MRM Conference four years ago, in which the co... more This is the sequel of a paper presented at the BEM/MRM Conference four years ago, in which the conventional, collocation boundary element method was reformulated by proposing a simple, however consistent derivation on the basis of the weighted-residuals statement. It was shown that the single-layer potential matrix G should be in general rectangular and satisfy some spectral properties (orthogonality to the space of unbalanced boundary traction forces) in the same way as the double-layer potential matrix H is orthogonal to rigid-body displacements, when modelling a finite elastic body. Moreover, a "subtle" improvement was proposed for the interpolation of traction forces, in the case of curved boundaries, which was meant to just simplify the numerical implementation. In the present paper, it is concluded that the proposed improvement is in fact a necessary one if strict consistency of the formulation is required and more emphatically if a consistent hypersingular formulation is to be implemented. It is also shown that the correct hypersingular formulation requires that the discontinuous parts (two free terms) of the matrix H be obtained independently from the matrix G. Motivation of the present developments was the application of the hybrid boundary element method to strain gradient elasticity, which only makes use of the matrix H together with its hypersingular counterpart. Although this paper is of a rather theoretical nature, a simple numerical example is shown to illustrate the necessity of the proposed improvements.
WIT transactions on modelling and simulation, Jun 28, 2011
The present developments combine the variationally-based, hybrid boundary element method with a c... more The present developments combine the variationally-based, hybrid boundary element method with a consistent formulation of the conventional, collocation boundary element method in order to establish a computationally less intensive procedure, although not necessarily less accurate, for large-scale, two-dimensional and three-dimensional problems of potential and elasticity, including timedependent phenomena. Both the double-layer and the single-layer potential matrices, H and G, whose evaluation usually requires dealing with singular and improper integrals, are obtained in an expedite way that circumvents almost any numerical integration-except for a few regular integrals in the case of H. A few numerical examples are shown to assess the applicability of the method, its computational effort and some convergence issues.
International Journal for Numerical Methods in Engineering, 2006
An algorithm is introduced for the inverse of a-matrix given as the truncated series A 0 − i A 1 ... more An algorithm is introduced for the inverse of a-matrix given as the truncated series A 0 − i A 1 − 2 A 2 + i 3 A 3 + 4 A 4 + • • • + O(n+1) with square coefficient matrices and singular leading term A 0. Moreover, A 1 may be conditionally singular and no restrictions are made for the remaining terms. The result is a-matrix given as a unique, truncated series of the same error order. Motivation for this problem is the evaluation of the frequency-dependent stiffness matrix of general boundary or macro-finite elements in the frame of a hybrid variational formulation that is based on a flexibility matrix F expressed as a truncated power series of the circular frequency .
WIT transactions on modelling and simulation, Mar 16, 2004
This paper introduces a brief, although insightful, comparative analysis of the conventional, col... more This paper introduces a brief, although insightful, comparative analysis of the conventional, collocation boundary element method (CBEM), the hybrid stress boundary element method (HSBEM) and the hybrid displacement boundary element method (HDBEM), not only investigating the mechanical properties of the resulting matrix equations, but mainly redefining a series of concepts in both HDBEM and CBEM that hadn't been properly considered by previous authors. This is not a review paper, but rather a theoretical investigation of three methods, with many physical considerations and some innovations that point toward mesh-reduced formulations adequate for the numerical modelling of complex problems, such as in the case of material non-homogeneity and timedependency.
WIT transactions on modelling and simulation, Aug 27, 2010
The collocation boundary element method is derived on the basis of the weighted-residuals stateme... more The collocation boundary element method is derived on the basis of the weighted-residuals statement. Only the static case is addressed, as it already involves all relevant conceptual issues. The present outline brings to discussion some relevant aspects and implementation issues of the method that should belong in any text book. It is shown that, if the boundary element method is consistently formulated, an inherent error term-related to arbitrary rigid-body displacements-is naturally taken into account and has no influence on the resultant matrix equation, with traction force parameters that are always in balance independently of mesh discretization. The constitutive matrices of the method-the single-layer and double-layer potential matrices G and H-present some spectral properties that are per se interesting but that also have applicability consequences. The matrix G is rectangular, if consistently obtained. For and adequately formulated problem, the solution of the resultant matrix equation is always possible (and unique) whether directly or approximated in terms of equivalent nodal forces. The effects of body forces, whenever transformable to boundary actions, may be expressed in terms of the boundary interpolation functions, which renders the final matrix equation more elegant and speeds up calculations in no detriment to accuracy. There is a novel proposition for the interpolation of traction forces along curved boundaries, with results that may be only slightly improved, as compared to the classical procedure, but that simplifies numerical computation and adds to the consistency of the method in terms of patch test assessments. The conceptual and numerical developments are illustrated by means of a few examples.
Cmes-computer Modeling in Engineering & Sciences, Oct 1, 2014
The present paper starts with Mindlin’s theory of the strain gradient elasticity, based on three ... more The present paper starts with Mindlin’s theory of the strain gradient elasticity, based on three additional constants for homogeneous materials (besides the Lamé’s constants), to arrive at a proposition made by Aifantis with just one additional parameter. Aifantis’characteristic material length g2, as it multiplies the Laplacian of the Cauchy stresses, may be seen as a penalty parameter to enforce interelement displacement gradient compatibility also in the case of a material in which the microstructure peculiarities are in principle not too relevant, but where high stress gradients occur. It is shown that the hybrid finite element formulation – as proposed by Pian and generalized by Dumont for finite and boundary elements – provides a natural conceptual framework to properly deal with the interelement compatibility of the normal displacement gradients, in which “corner nodes” are not an issue. Nonsingular fundamental solutions – domain interpolation functions – are presented for two-dimensional (2D) and three-dimensional (3D) problems, with the generation of families of finite elements that may be implemented in a straightforward way. Since the experimental data available in the technical literature are still scarce and the numerical results are in part questionable, consistency is assessed by means of patch tests and by investigating the spectral properties of the matrices derived for some 2D plane strain elements. The present developments, although of academical relevance, involve too many degrees of freedom to be considered for practical applications and are actually intended as a step toward a boundary-only implementation in terms of singular fundamental solutions.
WIT transactions on modelling and simulation, May 8, 2003
More than three decades ago, Przemieniecki [l] introduced a formulation for the free vibration an... more More than three decades ago, Przemieniecki [l] introduced a formulation for the free vibration analysis of bar and beam elements based on a power series of frequencies. Recently, this formulation was generalized for the analysis of the dynamic response of elastic systems submitted to arbitrary nodal loads as well as initial displacements [2]. Based on the mode-superposition method, a set of coupled, higher-order differential equations of motion is transformed into a set of uncoupled second order differential equations, which may be integrated by means of standard procedures. Motivation for this theoretical achievement is the hybrid boundary element method [3,4], as developed in [2] for time-dependent problems on the basis of a frequency-domain formulation, which, as a generalization of Pian's previous achievements for finite elements [ 5 ] , yields a stiffness matrix that requires only boundary integrals, for arbitrary domain shapes and any number of degrees of freedom. The use of higher-order frequency terms drastically improves numerical accuracy. The introduced modal assessment of the dynamic problem is applicable to any kind of finite element for which a generalized stiffness matrix is available [6, 71. The present paper is an attempt of consolidating this boundaryonly theoretical formulation, in which a series of particular cases are conceptually outlined and numerically assessed: Constrained and unconstrained structures; initial displacements and velocities as nodal values as well as prescribed domain fields (including rigid body movement); forced time-dependent displacements; self-weight and domain forces other than inertial forces; evaluation of results at internal points. Two academic examples for 2D problems of potential illustrate the formulation.
mitted to high gradients. numerical examples are given in terms of a patch test for irregular reg... more mitted to high gradients. numerical examples are given in terms of a patch test for irregular regions sub-method is preserved even with such spatially varying material property. Two tially varying thermal conductivity. Thus, the boundary-only feature of the tal solution is employed to model heat conduction in materials with exponen-problems of potential are derived. In particular, a recently developed fundamen-functionally graded materials. Several classes of fundamental solutions for the conceptual framework for applying the hybrid boundary element method to problems of elasticity and potential theory. This paper focuses on establishing The hybrid boundary element method has been successfully applied to various
Cmes-computer Modeling in Engineering & Sciences, Aug 1, 2011
... CMES, vol.78, no.2, pp.109-149, 2011 Generalized Westergaard Stress Functions as Fundamental ... more ... CMES, vol.78, no.2, pp.109-149, 2011 Generalized Westergaard Stress Functions as Fundamental Solutions ... Generalized Westergaard stress functions, as proposed by Tada, Ernst and Paris in 1993, are used as the problem's fundamental solution. ...
WIT transactions on modelling and simulation, Jun 11, 2013
The double-layer potential matrix H of the conventional, collocation boundary element method (CBE... more The double-layer potential matrix H of the conventional, collocation boundary element method (CBEM) is singular, as referred to a static problem in a bounded continuum. This means that no rigid body displacements can be transformed between two different reference systems-and that unbalanced forces are to be excluded from a consistent linear algebra contragradient transformation. The properties of H T may become quite informative, as they reflect the topology (concavities, notches, cracks, holes) of the discretized domain as well as material non-homogeneities, which comes from the fact that local stress gradients can be represented by fundamental solutions only in a global sense. Symmetries and antisymmetries are also evidenced in N (H T) as well as the spectral properties related to simple polynomial solutions that one may propose as patch tests. In the usual implementations of the CBEM with real fundamental solutions, all eigenvalues λ of H are real, λ ∈ R, 0 ≥ λ < 1 for a bounded domain. This means that H is a contraction-a paramount mechanical feature that comes up naturally in the frame of a virtual work investigation of Kelvin's (singular) fundamental solution and is resorted to in a simplified variational implementation of the boundary element method.
WIT transactions on engineering sciences, Aug 2, 2022
This paper is part of a research work to implement, test, and apply a novel numerical tool that c... more This paper is part of a research work to implement, test, and apply a novel numerical tool that can simulate on a personal computer and in just a few minutes a problem of potential or elasticity with up to tens of millions of degrees of freedom. The first author's group has already developed their own version of the fast multipole method (FMM) for two-dimensional problems, which relies on a consistent construction of the single-layer potential matrix of the collocation boundary element method so that ultimately only polynomial terms (as for the double-layer potential matrix) are required to be integrated along generally curved segments related to a given field expansion pole. The core of the present paper is the mathematical assessment of the double expansions needed in the 3D FMM. The 3D implementation is combined with a particular formulation for linear triangle elements in which all integrations for adjacent source point and boundary element are carried out analytically. As a result, numerical approximations are due exclusively to the FMM series truncations. This allows isolating and testing truncation errors incurred in the series expansions and thus for the first time properly assessing the mathematical features of the FMM, as illustrated by means of two examples. Adaptive numerical quadratures as well as the complete solution of a mixed boundary problem using a GMRES solver, for instance, are just additional tasks and, although already implemented, are not reported herein.
A boundary integral implementation generally involves the evaluation of singular and quasi-singul... more A boundary integral implementation generally involves the evaluation of singular and quasi-singular integrals. Dealing with them requires elaborated codes, which quite frequently results in prohibitive computational effort as the price of adequate numerical accuracy. In the frame of a research line in progress at PUC-Rio, a simple and accurate scheme has been developed for the treatment of general singular linear integrals. It is now being implemented for double integrals. This technique makes use of fixed abscissas, as taken from Gauss-Legendre quadrature, with different weight sets calculated for different singularities — a procedure that demands low computational effort. This feature allows the development of robust codes, in which integration demands low computational effort and yields high precision. The core of the present contribution is the introduction of an adequate scheme for the global-to-local transformation of the coordinates of a given singularity pole. This transformation may be used in connection with polar coordinates, as singular integrals are usually dealt with. In the paper, however, it is an essential feature of the unified integration procedure the authors are proposing. The technique is straightforward to implement, independently of the degree of singularity or quasi-singularity.
Lecture notes in computational science and engineering, 2017
We present a simple, PDE-based proof of the result [17] by M. Johnson that the error estimates of... more We present a simple, PDE-based proof of the result [17] by M. Johnson that the error estimates of J. Duchon [11] for thin plate spline interpolation can be improved by h 1/2. We illustrate that H-matrix techniques can successfully be employed to solve very large thin plate spline interpolation problems.
WIT transactions on modelling and simulation, Sep 8, 2014
This is the sequel of a paper presented at the BEM/MRM Conference four years ago, in which the co... more This is the sequel of a paper presented at the BEM/MRM Conference four years ago, in which the conventional, collocation boundary element method was reformulated by proposing a simple, however consistent derivation on the basis of the weighted-residuals statement. It was shown that the single-layer potential matrix G should be in general rectangular and satisfy some spectral properties (orthogonality to the space of unbalanced boundary traction forces) in the same way as the double-layer potential matrix H is orthogonal to rigid-body displacements, when modelling a finite elastic body. Moreover, a "subtle" improvement was proposed for the interpolation of traction forces, in the case of curved boundaries, which was meant to just simplify the numerical implementation. In the present paper, it is concluded that the proposed improvement is in fact a necessary one if strict consistency of the formulation is required and more emphatically if a consistent hypersingular formulation is to be implemented. It is also shown that the correct hypersingular formulation requires that the discontinuous parts (two free terms) of the matrix H be obtained independently from the matrix G. Motivation of the present developments was the application of the hybrid boundary element method to strain gradient elasticity, which only makes use of the matrix H together with its hypersingular counterpart. Although this paper is of a rather theoretical nature, a simple numerical example is shown to illustrate the necessity of the proposed improvements.
WIT transactions on modelling and simulation, Jun 28, 2011
The present developments combine the variationally-based, hybrid boundary element method with a c... more The present developments combine the variationally-based, hybrid boundary element method with a consistent formulation of the conventional, collocation boundary element method in order to establish a computationally less intensive procedure, although not necessarily less accurate, for large-scale, two-dimensional and three-dimensional problems of potential and elasticity, including timedependent phenomena. Both the double-layer and the single-layer potential matrices, H and G, whose evaluation usually requires dealing with singular and improper integrals, are obtained in an expedite way that circumvents almost any numerical integration-except for a few regular integrals in the case of H. A few numerical examples are shown to assess the applicability of the method, its computational effort and some convergence issues.
International Journal for Numerical Methods in Engineering, 2006
An algorithm is introduced for the inverse of a-matrix given as the truncated series A 0 − i A 1 ... more An algorithm is introduced for the inverse of a-matrix given as the truncated series A 0 − i A 1 − 2 A 2 + i 3 A 3 + 4 A 4 + • • • + O(n+1) with square coefficient matrices and singular leading term A 0. Moreover, A 1 may be conditionally singular and no restrictions are made for the remaining terms. The result is a-matrix given as a unique, truncated series of the same error order. Motivation for this problem is the evaluation of the frequency-dependent stiffness matrix of general boundary or macro-finite elements in the frame of a hybrid variational formulation that is based on a flexibility matrix F expressed as a truncated power series of the circular frequency .
WIT transactions on modelling and simulation, Mar 16, 2004
This paper introduces a brief, although insightful, comparative analysis of the conventional, col... more This paper introduces a brief, although insightful, comparative analysis of the conventional, collocation boundary element method (CBEM), the hybrid stress boundary element method (HSBEM) and the hybrid displacement boundary element method (HDBEM), not only investigating the mechanical properties of the resulting matrix equations, but mainly redefining a series of concepts in both HDBEM and CBEM that hadn't been properly considered by previous authors. This is not a review paper, but rather a theoretical investigation of three methods, with many physical considerations and some innovations that point toward mesh-reduced formulations adequate for the numerical modelling of complex problems, such as in the case of material non-homogeneity and timedependency.
WIT transactions on modelling and simulation, Aug 27, 2010
The collocation boundary element method is derived on the basis of the weighted-residuals stateme... more The collocation boundary element method is derived on the basis of the weighted-residuals statement. Only the static case is addressed, as it already involves all relevant conceptual issues. The present outline brings to discussion some relevant aspects and implementation issues of the method that should belong in any text book. It is shown that, if the boundary element method is consistently formulated, an inherent error term-related to arbitrary rigid-body displacements-is naturally taken into account and has no influence on the resultant matrix equation, with traction force parameters that are always in balance independently of mesh discretization. The constitutive matrices of the method-the single-layer and double-layer potential matrices G and H-present some spectral properties that are per se interesting but that also have applicability consequences. The matrix G is rectangular, if consistently obtained. For and adequately formulated problem, the solution of the resultant matrix equation is always possible (and unique) whether directly or approximated in terms of equivalent nodal forces. The effects of body forces, whenever transformable to boundary actions, may be expressed in terms of the boundary interpolation functions, which renders the final matrix equation more elegant and speeds up calculations in no detriment to accuracy. There is a novel proposition for the interpolation of traction forces along curved boundaries, with results that may be only slightly improved, as compared to the classical procedure, but that simplifies numerical computation and adds to the consistency of the method in terms of patch test assessments. The conceptual and numerical developments are illustrated by means of a few examples.
Cmes-computer Modeling in Engineering & Sciences, Oct 1, 2014
The present paper starts with Mindlin’s theory of the strain gradient elasticity, based on three ... more The present paper starts with Mindlin’s theory of the strain gradient elasticity, based on three additional constants for homogeneous materials (besides the Lamé’s constants), to arrive at a proposition made by Aifantis with just one additional parameter. Aifantis’characteristic material length g2, as it multiplies the Laplacian of the Cauchy stresses, may be seen as a penalty parameter to enforce interelement displacement gradient compatibility also in the case of a material in which the microstructure peculiarities are in principle not too relevant, but where high stress gradients occur. It is shown that the hybrid finite element formulation – as proposed by Pian and generalized by Dumont for finite and boundary elements – provides a natural conceptual framework to properly deal with the interelement compatibility of the normal displacement gradients, in which “corner nodes” are not an issue. Nonsingular fundamental solutions – domain interpolation functions – are presented for two-dimensional (2D) and three-dimensional (3D) problems, with the generation of families of finite elements that may be implemented in a straightforward way. Since the experimental data available in the technical literature are still scarce and the numerical results are in part questionable, consistency is assessed by means of patch tests and by investigating the spectral properties of the matrices derived for some 2D plane strain elements. The present developments, although of academical relevance, involve too many degrees of freedom to be considered for practical applications and are actually intended as a step toward a boundary-only implementation in terms of singular fundamental solutions.
WIT transactions on modelling and simulation, May 8, 2003
More than three decades ago, Przemieniecki [l] introduced a formulation for the free vibration an... more More than three decades ago, Przemieniecki [l] introduced a formulation for the free vibration analysis of bar and beam elements based on a power series of frequencies. Recently, this formulation was generalized for the analysis of the dynamic response of elastic systems submitted to arbitrary nodal loads as well as initial displacements [2]. Based on the mode-superposition method, a set of coupled, higher-order differential equations of motion is transformed into a set of uncoupled second order differential equations, which may be integrated by means of standard procedures. Motivation for this theoretical achievement is the hybrid boundary element method [3,4], as developed in [2] for time-dependent problems on the basis of a frequency-domain formulation, which, as a generalization of Pian's previous achievements for finite elements [ 5 ] , yields a stiffness matrix that requires only boundary integrals, for arbitrary domain shapes and any number of degrees of freedom. The use of higher-order frequency terms drastically improves numerical accuracy. The introduced modal assessment of the dynamic problem is applicable to any kind of finite element for which a generalized stiffness matrix is available [6, 71. The present paper is an attempt of consolidating this boundaryonly theoretical formulation, in which a series of particular cases are conceptually outlined and numerically assessed: Constrained and unconstrained structures; initial displacements and velocities as nodal values as well as prescribed domain fields (including rigid body movement); forced time-dependent displacements; self-weight and domain forces other than inertial forces; evaluation of results at internal points. Two academic examples for 2D problems of potential illustrate the formulation.
mitted to high gradients. numerical examples are given in terms of a patch test for irregular reg... more mitted to high gradients. numerical examples are given in terms of a patch test for irregular regions sub-method is preserved even with such spatially varying material property. Two tially varying thermal conductivity. Thus, the boundary-only feature of the tal solution is employed to model heat conduction in materials with exponen-problems of potential are derived. In particular, a recently developed fundamen-functionally graded materials. Several classes of fundamental solutions for the conceptual framework for applying the hybrid boundary element method to problems of elasticity and potential theory. This paper focuses on establishing The hybrid boundary element method has been successfully applied to various
Cmes-computer Modeling in Engineering & Sciences, Aug 1, 2011
... CMES, vol.78, no.2, pp.109-149, 2011 Generalized Westergaard Stress Functions as Fundamental ... more ... CMES, vol.78, no.2, pp.109-149, 2011 Generalized Westergaard Stress Functions as Fundamental Solutions ... Generalized Westergaard stress functions, as proposed by Tada, Ernst and Paris in 1993, are used as the problem's fundamental solution. ...
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