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Y C Shiah - One of the best experts on this subject based on the ideXlab platform.

  • analytical transformation of the volume integral in the Boundary integral equation for 3d anisotropic elastostatics involving body force
    Computer Methods in Applied Mechanics and Engineering, 2014
    Co-Authors: Y C Shiah
    Abstract:

    Abstract In the Boundary element method (BEM), it is well known that the presence of body force shall give rise to an additional volume integral that conventionally requires domain discretization for numerical computations. To restore the BEM’s distinctive notion of Boundary discretization, the present work analytically transforms the volume integral to surface ones for the body-force effect in the 3D anisotropic elasticity. On applying Green’s Theorem, new fundamental Solutions with explicit forms of Fourier series are introduced to facilitate the volume-to-surface transformation. The coefficients of the Fourier-series representations are determined by solving a banded matrix formulated from integrations of the constrained equation. Of no doubt, such an approach has fully restored the Boundary element method as a truly Boundary Solution technique for analyzing 3D anisotropic elasticity involving body force. At the end, numerical verifications of the volume-to-surface integral transformation are presented. Also, such an approach has been implemented in an existing BEM code. For demonstrating the implementation, numerical examples are presented with comparisons with ANSYS analysis. To the author’s knowledge, this is the first work in the open literature that reports the successful transformation for 3D anisotropic elasticity.

  • direct volume to surface integral transformation for 2d bem analysis of anisotropic thermoelasticity
    Cmes-computer Modeling in Engineering & Sciences, 2014
    Co-Authors: Y C Shiah, M H Aliabadi, Chyanbin Hwu
    Abstract:

    As has been well documented for the Boundary element method (BEM), a volume integral is present in the integral equation for thermoelastic analysis. Any attempt to directly integrate the integral shall inevitably involve internal discretisation that will destroy the BEM's notion as a true Boundary Solution technique. Among the schemes to overcome this difficulty, the exact transformation approach is the most elegant since neither further approximation nor internal treatments are involved. Such transformation for 2D anisotropic thermoelasticity has been achieved by Shiah and Tan [1] with the aid of domain mapping. This paper revisits this problem and presents a modified transformation process for 2D anisotropic thermoelasticity, where no domain distortion is involved. Being defined in the original Cartesian coordinate system, the volume integral is analytically transformed to the Boundary, being derived using the Stroh formulism. This transformation is favorable especially when the corresponding anisotropic field is directly calculated using the anisotropic Green's function without resorting to domain mapping. At the end, numerical examples are provided to show the validity of such transformation.

  • anisotropic heat conduction involving internal arbitrary volume heat generation rate
    International Communications in Heat and Mass Transfer, 2002
    Co-Authors: Y C Shiah
    Abstract:

    In this article, the problem of anisotropic heat conduction involving internal arbitrary volume heat generation rate is solved by the Boundary element method. In the direct formulation for the Boundary element analysis ofheat conduction problems, the presence of a volume heat source will give rise to a domain integral, which conventionally demands internal cell discretisation throughout the whole domain. However, this domain discretisation will destroy the distinctive feature of the Boundary element method as a Boundary Solution technique. In this paper, the multiple reciprocity method in conjunction with the standard characteristics method is employed to exactly transform the domain integral into a series of Boundary ones. The distribution of heat generation in a volume heat source could be in an arbitrary form of a continuous function. After the anisotropic heat conduction problem is iteratively solved in the mapped plane, the obtained numerical Solution is thereafter interpolated and transformed back to the one in the physical domain.

  • exact Boundary integral transformation of the thermoelastic domain integral in bem for general 2d anisotropic elasticity
    Computational Mechanics, 1999
    Co-Authors: Y C Shiah, C L Tan
    Abstract:

    In the direct formulation of the Boundary element method, body-force and thermal loads manifest themselves as additional volume integral terms in the Boundary integral equation. The exact transformation of the volume integral associated with body-force loading into surface ones for two-dimensional elastostatics in general anisotropy, has only very recently been achieved. This paper extends the work to treat two-dimensional thermoelastic problems which, unlike in isotropic elasticity, pose additional complications in the formulation. The success of the exact volume-to-surface integral transformation and its implementation is illustrated with three examples. The present study restores the application of BEM to two-dimensional anisotropic elastostatics as a truly Boundary Solution technique even when thermal effects are involved.

Fred F Afagh - One of the best experts on this subject based on the ideXlab platform.

  • treatment of body force volume integrals in bem by exact transformation for 2 d anisotropic elasticity
    International Journal for Numerical Methods in Engineering, 1997
    Co-Authors: J J Zhang, C L Tan, Fred F Afagh
    Abstract:

    In the Boundary Element Method (BEM) based on the direct formulation, body-force effects manifest themselves as an additional volume integral term in the Boundary Integral Equation (BIE). The numerical Solution of the integral equation with this term destroys the notion of the BEM as atruly Boundary Solution method. This paper discusses the treatment of this volume integral for two-dimensional anisotropic elasticity with body-forces present. The analytical basis for transforming this integral exactly into Boundary ones is presented for geometrically convex regions. This restores the application of the BEM to such problems as a truly Boundary Solution technique. Numerical examples are presented to demonstrate the veracity of the transformation and implementation. © 1997 by John Wiley & Sons, Ltd.

  • a general exact transformation of body force volume integral in bem for 2d anisotropic elasticity
    Computational Mechanics, 1996
    Co-Authors: J J Zhang, C L Tan, Fred F Afagh
    Abstract:

    In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to Boundary ones. This restores the BEM as a truly Boundary Solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.

C L Tan - One of the best experts on this subject based on the ideXlab platform.

  • exact Boundary integral transformation of the thermoelastic domain integral in bem for general 2d anisotropic elasticity
    Computational Mechanics, 1999
    Co-Authors: Y C Shiah, C L Tan
    Abstract:

    In the direct formulation of the Boundary element method, body-force and thermal loads manifest themselves as additional volume integral terms in the Boundary integral equation. The exact transformation of the volume integral associated with body-force loading into surface ones for two-dimensional elastostatics in general anisotropy, has only very recently been achieved. This paper extends the work to treat two-dimensional thermoelastic problems which, unlike in isotropic elasticity, pose additional complications in the formulation. The success of the exact volume-to-surface integral transformation and its implementation is illustrated with three examples. The present study restores the application of BEM to two-dimensional anisotropic elastostatics as a truly Boundary Solution technique even when thermal effects are involved.

  • treatment of body force volume integrals in bem by exact transformation for 2 d anisotropic elasticity
    International Journal for Numerical Methods in Engineering, 1997
    Co-Authors: J J Zhang, C L Tan, Fred F Afagh
    Abstract:

    In the Boundary Element Method (BEM) based on the direct formulation, body-force effects manifest themselves as an additional volume integral term in the Boundary Integral Equation (BIE). The numerical Solution of the integral equation with this term destroys the notion of the BEM as atruly Boundary Solution method. This paper discusses the treatment of this volume integral for two-dimensional anisotropic elasticity with body-forces present. The analytical basis for transforming this integral exactly into Boundary ones is presented for geometrically convex regions. This restores the application of the BEM to such problems as a truly Boundary Solution technique. Numerical examples are presented to demonstrate the veracity of the transformation and implementation. © 1997 by John Wiley & Sons, Ltd.

  • a general exact transformation of body force volume integral in bem for 2d anisotropic elasticity
    Computational Mechanics, 1996
    Co-Authors: J J Zhang, C L Tan, Fred F Afagh
    Abstract:

    In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to Boundary ones. This restores the BEM as a truly Boundary Solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.

J J Zhang - One of the best experts on this subject based on the ideXlab platform.

  • treatment of body force volume integrals in bem by exact transformation for 2 d anisotropic elasticity
    International Journal for Numerical Methods in Engineering, 1997
    Co-Authors: J J Zhang, C L Tan, Fred F Afagh
    Abstract:

    In the Boundary Element Method (BEM) based on the direct formulation, body-force effects manifest themselves as an additional volume integral term in the Boundary Integral Equation (BIE). The numerical Solution of the integral equation with this term destroys the notion of the BEM as atruly Boundary Solution method. This paper discusses the treatment of this volume integral for two-dimensional anisotropic elasticity with body-forces present. The analytical basis for transforming this integral exactly into Boundary ones is presented for geometrically convex regions. This restores the application of the BEM to such problems as a truly Boundary Solution technique. Numerical examples are presented to demonstrate the veracity of the transformation and implementation. © 1997 by John Wiley & Sons, Ltd.

  • a general exact transformation of body force volume integral in bem for 2d anisotropic elasticity
    Computational Mechanics, 1996
    Co-Authors: J J Zhang, C L Tan, Fred F Afagh
    Abstract:

    In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to Boundary ones. This restores the BEM as a truly Boundary Solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.

Richard Schoen - One of the best experts on this subject based on the ideXlab platform.

  • the first steklov eigenvalue conformal geometry and minimal surfaces
    Advances in Mathematics, 2011
    Co-Authors: Ailana Fraser, Richard Schoen
    Abstract:

    We consider the relationship of the geometry of compact Riemannian manifolds with Boundary to the first nonzero eigenvalue σ1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k Boundary components we obtain the upper bound σ1L(∂Σ)⩽2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a Solution of the free Boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ1(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the Boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free Boundary Solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free Boundary Solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free Boundary Solutions in the two-dimensional case.

  • the first steklov eigenvalue conformal geometry and minimal surfaces
    arXiv: Differential Geometry, 2009
    Co-Authors: Ailana Fraser, Richard Schoen
    Abstract:

    We consider the relationship of the geometry of compact Riemannian manifolds with Boundary to the first nonzero eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Sigma with genus gamma and k Boundary components we obtain the upper bound sigma_1L(\partial \Sigma) \leq 2(2gamma+k)\pi. We attempt to find the best constant in this inequality for annular surfaces (gamma=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a Solution of the free Boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that $\sigma_1(\sig)L(\p\Sigma)$ is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the Boundary. We prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free Boundary Solution in two dimensions has area at least \pi.