In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic ... more In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic problem. The present quaternion-valued BEM is developed by discretizing the quaternion-valued boundary integral equation (BIE). The quaternionvalued BIE can be seen as an extension of the generalized complex variable BIE in 3-D space. In other words, quaternion algebra is an extension of the complex variable in 3-D space. To derive quaternion-valued BIEs, the quaternion-valued Stokes' theorem is utilized. The quaternion-valued BIEs are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). An analytical scheme is developed to evaluate the CPV by introducing a simple quaternion-valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called "numerical boundary layer" phenomena appear and are remedied by using a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. It is noted that all derivations are done in quaternion algebra. In addition, quaternion algebra is more flexible than vector algebra in solving some 3-D problems from the point view of algebraic space. In deriving BIEs for exterior fields, the conditions at infinity for the quaternion-valued functions are carefully examined. Later, a magnetic sphere in a uniform magnetic field is considered. This problem is a magnetostatic problem of coupled exterior and interior magnetostatic fields. Finally, we apply the present approach to solve the magnetostatic problem. By comparing with exact solutions, the validity of the present approach is checked.
Journal of Engineering Mechanics-asce, Jun 1, 1988
In this paper, we derive the integral equations of elasticity, which may be considered to be a ve... more In this paper, we derive the integral equations of elasticity, which may be considered to be a very general formulation for solutions of (cracked and uncracked) elasticity problems. The formulation is general enough to be a starting point for an analytical study or for a numerical treatment. The theory can be developed either by utilizing Betti's law or the weighted sesidual method, or directly resorting to physical meaning, as in the potential theory. To show that the results of the derivations are consistent with one another, we also prove four lemmas of the properties of the kernel functions. The derivations are continued by applying two commutative operations, traction and trace, leading naturally to the concept of Hadamard principal value. Consequently, singularity, often present in problems involving geometry degeneracy, causes no particular difficulties. Finally, we employ two examples to demonstrate the usefulness of the resulting dual boundary integral equations in both analytical and numerical solution procedures.
Engineering Analysis With Boundary Elements, May 1, 2015
Theory of complex variables is a very powerful mathematical technique for solving two-dimensional... more Theory of complex variables is a very powerful mathematical technique for solving two-dimensional problems satisfying the Laplace equation. On the basis of the conventional Cauchy integral formula, the conventional complex variable boundary integral equation (CVBIE) can be constructed. The limitation is that the conventional CVBIE is only suitable for holomorphic (analytic) functions, however. To solve for a complex-valued harmonic-function pair without satisfying the Cauchy-Riemann equations, we propose a new boundary element method (BEM) based on the general Cauchy integral formula. The general Cauchy integral formula is derived by using the Borel-Pompeiu formula. The difference between the present CVBIE and the conventional CVBIE is that the former one has two boundary integrals instead of only one boundary integral in the latter one. When the unknown field is a holomorphic function, the present CVBIE can be reduced to the conventional CVBIE. Therefore, the conventional Cauchy integral formula can be viewed as a special case applicable to a holomorphic function. To examine the present CVBIE, we consider several torsion problems in this paper since the two shear stress fields satisfy the Laplace equation but do not satisfy the Cauchy-Riemann equations. Using the present CVBIE, we can directly solve the stress fields and the torsional rigidity simultaneously. Finally, several examples, including a circular bar containing an eccentric inclusion (with dissimilar materials) or hole, a circular bar, elliptical bar, equilateral triangular bar, rectangular bar, asteroid bar and circular bar with keyway, were demonstrated to check the validity of the present method.
In this paper, the dual integral representation in series forms or briefly the dual series repres... more In this paper, the dual integral representation in series forms or briefly the dual series representation for transient vibration problems is derived using the concept of modal dynamics. The four kernel functions in this representation are in degenerate series forms which display symmetric and transpose symmetric properties. Assembling the four kernel functions, the total symmetry can be obtained. The potentials induced by the kernels are expressed in terms of uniformly convergent, pointwise convergent, and divergent series for modal dynamic problems investigated in this paper, instead of continuous, discontinuous, and pseudo-continuous functions in the form of integrals with different degrees of singularity for static and direct transient problems. The technique used for the extraction of the principal values of the singular integrals is herein transformed to that of summability of the series or Stokes' transformation. The Cesaro sum technique is employed not only to treat poin...
Developed herein is an analysis procedure based on closed-form solutions to elastoplastic bilinea... more Developed herein is an analysis procedure based on closed-form solutions to elastoplastic bilinear model of building structures accounted for different stiffnesses and yielding forces in different directions and rotated yield ellipses in different floor levels due to the layout of buildings and the complexity of structural members. The seismic design often considers earthquake forces on multiple floor levels but usually only in a single direction. However, in reality, the direction of the earthquake is not limited to one particular direction. Therefore, studying the influence of a two-way, furthermore multi-dimensional, earthquake on buildings is of great value. To estimate the total seismic demand on inelastic building structures subjected to multi-dimensional loading, this paper aims to find closed-form solution responses to an input rectilinear force path for the elastoplastic bilinear model of Hong and Liu (1999) which already has available closed-form solution responses to an i...
In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic ... more In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic problem. The present quaternion-valued BEM is developed by discretizing the quaternion-valued boundary integral equation (BIE). The quaternionvalued BIE can be seen as an extension of the generalized complex variable BIE in 3-D space. In other words, quaternion algebra is an extension of the complex variable in 3-D space. To derive quaternion-valued BIEs, the quaternion-valued Stokes' theorem is utilized. The quaternion-valued BIEs are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). An analytical scheme is developed to evaluate the CPV by introducing a simple quaternion-valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called "numerical boundary layer" phenomena appear and are remedied by using a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. It is noted that all derivations are done in quaternion algebra. In addition, quaternion algebra is more flexible than vector algebra in solving some 3-D problems from the point view of algebraic space. In deriving BIEs for exterior fields, the conditions at infinity for the quaternion-valued functions are carefully examined. Later, a magnetic sphere in a uniform magnetic field is considered. This problem is a magnetostatic problem of coupled exterior and interior magnetostatic fields. Finally, we apply the present approach to solve the magnetostatic problem. By comparing with exact solutions, the validity of the present approach is checked.
Journal of Engineering Mechanics-asce, Jun 1, 1988
In this paper, we derive the integral equations of elasticity, which may be considered to be a ve... more In this paper, we derive the integral equations of elasticity, which may be considered to be a very general formulation for solutions of (cracked and uncracked) elasticity problems. The formulation is general enough to be a starting point for an analytical study or for a numerical treatment. The theory can be developed either by utilizing Betti's law or the weighted sesidual method, or directly resorting to physical meaning, as in the potential theory. To show that the results of the derivations are consistent with one another, we also prove four lemmas of the properties of the kernel functions. The derivations are continued by applying two commutative operations, traction and trace, leading naturally to the concept of Hadamard principal value. Consequently, singularity, often present in problems involving geometry degeneracy, causes no particular difficulties. Finally, we employ two examples to demonstrate the usefulness of the resulting dual boundary integral equations in both analytical and numerical solution procedures.
Engineering Analysis With Boundary Elements, May 1, 2015
Theory of complex variables is a very powerful mathematical technique for solving two-dimensional... more Theory of complex variables is a very powerful mathematical technique for solving two-dimensional problems satisfying the Laplace equation. On the basis of the conventional Cauchy integral formula, the conventional complex variable boundary integral equation (CVBIE) can be constructed. The limitation is that the conventional CVBIE is only suitable for holomorphic (analytic) functions, however. To solve for a complex-valued harmonic-function pair without satisfying the Cauchy-Riemann equations, we propose a new boundary element method (BEM) based on the general Cauchy integral formula. The general Cauchy integral formula is derived by using the Borel-Pompeiu formula. The difference between the present CVBIE and the conventional CVBIE is that the former one has two boundary integrals instead of only one boundary integral in the latter one. When the unknown field is a holomorphic function, the present CVBIE can be reduced to the conventional CVBIE. Therefore, the conventional Cauchy integral formula can be viewed as a special case applicable to a holomorphic function. To examine the present CVBIE, we consider several torsion problems in this paper since the two shear stress fields satisfy the Laplace equation but do not satisfy the Cauchy-Riemann equations. Using the present CVBIE, we can directly solve the stress fields and the torsional rigidity simultaneously. Finally, several examples, including a circular bar containing an eccentric inclusion (with dissimilar materials) or hole, a circular bar, elliptical bar, equilateral triangular bar, rectangular bar, asteroid bar and circular bar with keyway, were demonstrated to check the validity of the present method.
In this paper, the dual integral representation in series forms or briefly the dual series repres... more In this paper, the dual integral representation in series forms or briefly the dual series representation for transient vibration problems is derived using the concept of modal dynamics. The four kernel functions in this representation are in degenerate series forms which display symmetric and transpose symmetric properties. Assembling the four kernel functions, the total symmetry can be obtained. The potentials induced by the kernels are expressed in terms of uniformly convergent, pointwise convergent, and divergent series for modal dynamic problems investigated in this paper, instead of continuous, discontinuous, and pseudo-continuous functions in the form of integrals with different degrees of singularity for static and direct transient problems. The technique used for the extraction of the principal values of the singular integrals is herein transformed to that of summability of the series or Stokes' transformation. The Cesaro sum technique is employed not only to treat poin...
Developed herein is an analysis procedure based on closed-form solutions to elastoplastic bilinea... more Developed herein is an analysis procedure based on closed-form solutions to elastoplastic bilinear model of building structures accounted for different stiffnesses and yielding forces in different directions and rotated yield ellipses in different floor levels due to the layout of buildings and the complexity of structural members. The seismic design often considers earthquake forces on multiple floor levels but usually only in a single direction. However, in reality, the direction of the earthquake is not limited to one particular direction. Therefore, studying the influence of a two-way, furthermore multi-dimensional, earthquake on buildings is of great value. To estimate the total seismic demand on inelastic building structures subjected to multi-dimensional loading, this paper aims to find closed-form solution responses to an input rectilinear force path for the elastoplastic bilinear model of Hong and Liu (1999) which already has available closed-form solution responses to an i...
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