Closed-form Fundamental Solution Research Articles

Unsteady two-temperature heat transport in mass-in-mass chains.

We investigate the unsteady heat (energy) transport in an infinite mass-in-mass chain with a given initial temperature profile. The chain consists of two sublattices: the β-Fermi-Pasta-Ulam-Tsingou (FPUT) chain and oscillators (of a different mass) connected to each FPUT particle. Initial conditions are such that initial kinetic temperatures of the FPUT particles and the oscillators are equal. Using the harmonic theory, we analytically describe evolution of these two temperatures in the ballistic regime. In particular, we derive a closed-form fundamental solution and solution for a sinusoidal initial temperature profile in the case when the oscillators are significantly lighter than the FPUT particles. The harmonic theory predicts that during the heat transfer the temperatures of sublattices are significantly different, while initially and finally (at large times) they are equal. This may look like an artifact of the harmonic approximation, but we show that it is not the case. Two distinct temperatures are also observed in the anharmonic case, even when the heat transport regime is no longer quasiballistic. We show that the value of the nonlinearity coefficient required to equalize the temperatures strongly depends on the particle mass ratio. If the oscillators are much lighter than the FPUT particles, then a fairly strong nonlinearity is required for the equalization.

Analytical solution for potential flow across two circular cylinders using the BIE in conjunction with degenerate kernels of bipolar coordinates

In this paper, we employ the bipolar degenerate-kernel in the boundary integral equation (BIE) to derive the analytical solution of potential flow across two identical circular cylinders. Analytical results are compared with those in Lebedev et al.. Discussions of the difference between our results and Lebedev’s solution are addressed. Velocity distribution is also plotted. Both analytical solution and velocity plots are given. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using the bipolar coordinates for a case of two cylinders. The orientation, or so called the angle of attack is considered. The result shows that the degenerate-kernel approach can be an alternative tool for analytically solving some boundary value problems, although the complex variables is always employed. The extension to 3-D problems is promising and straightforward once the degenerate kernel is available since complex variable may have difficulty.

Applications of degenerate kernels to potential flow across circular, elliptical cylinders and a thin airfoil

In this paper, we employ the boundary integral equation (BIE) to analytically derive the solution of potential flow across a circular cylinder, an elliptical cylinder and a thin airfoil. It is found that the symmetric or antisymmetric potential flow problem can be solved by using the UT or the LM equation alone. Both analytical and numerical approaches were considered. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using polar and elliptical coordinates for a circular cylinder and an elliptical cylinder, respectively. The role of the dual BEM is also examined by showing the singular vectors of influence matrix. Besides, either the singular or the hypersingular BIE are respectively employed alone to solve the problem of symmetric and anti-symmetric flow field. Analytical derivation as well as the BEM is demonstrated. The orientation of the ellipse, as well as the angle of attack for the thin airfoil, is considered. The result shows that the degenerate kernel can be an alternative tool for solving some BVPs, although the complex variable is always employed. Once the degenerate kernel for the closed-form fundamental solution is available, the BIE is nothing more than the linear algebra and an analytical study is possible. The extension to 3-D object is promising once the degenerate kernel is available.

On the stress concentration factor of circular/elliptic hole and rigid inclusion under the remote anti-plane shear by using degenerate kernels

The stress concentration factor (SCF) along the boundary of a hole and a rigid inclusion in an infinite isotropic solid under the anti-plane shear is revisited by using degenerate kernels in the boundary integral equation (BIE) although this result was obtained by invoking the extended circle theorem of Milne-Thomson as well as the complex variable approach. The degenerate kernel of series form for the closed-form fundamental solution is used for the circle and the ellipse in terms of polar and elliptic coordinates, respectively. The slender ratio of the ellipse and the orientation are two parameters for our study. The strain energy density along the boundary is increased or decreased due to the different types of loading and various aspect ratios of the ellipse. An analytical solution for the SCF is then derived for any orientation of the ellipse relative to the applied load. The reciprocal relation for the SCF between a hole and a rigid inclusion with respect to different loading is also addressed. Besides, this analytical derivation can clearly show the appearing mechanism why the BEM/BIEM suffers the degenerate scale in the rigid inclusion.

Analytical and numerical studies for solving Steklov eigenproblems by using the boundary integral equation method/boundary element method

The theory of boundary eigensolutions is developed for boundary value problems. It is general for boundary value problem. Steklov-Poincaré operator maps the values of a boundary condition of the solution of the Laplace equation in a domain to the values of another boundary condition. The eigenvalue is imbedded in the Dirichlet to Neumann (DtN) map. The DtN operator is called the Steklov operator. In this paper, we study the Steklov eigenproblems by using the dual boundary element method/boundary integral equation method (BEM/BIEM). First, we consider a circular domain. To analytically derive the eigensolution of the above shape, the closed-form fundamental solution of the 2D Laplace equation, ln(r), is expanded into degenerate kernel by using the polar coordinate system. After the boundary element discretization of the BIE for the Steklov eigenproblem, it can be transformed to a standard linear eigenequation. Problems can be effectively solved by using the dual BEM. Finally, we consider the annulus. Not only the Steklov problem but also the mixed Steklov eigenproblem for an annular domain has been considered.

Analytical derivation and numerical experiment of degenerate scale by using the degenerate kernel of the bipolar coordinates

Degenerate scales of an eccentric annulus and an infinite plane with two identical circular holes in the boundary integral equation method (BIEM) are analytically derived and numerically implemented in this paper. To analytically study the degenerate scale of the BIE, the closed-form fundamental solution of the two-dimensional Laplace equation, ln r, is expanded by a degenerate (separate) kernel in terms of the bipolar coordinates. It is proved that unit radius of the outer circle dominates the degenerate scale of eccentric annulus. An analytical formula of degenerate scale for the infinite plane with two identical circular boundaries was also derived at the first time. In addition, null fields of the domain and complementary domain for the ordinary and degenerate scales are both shown, respectively. Finally, comparison with available results and the BEM data are well done.

Acoustic wave propagation in horizontally variable shallow water

A unique normal mode solution which approximates sound speed variation with depth by a sequence of isovelocity layers is verified for a sample of range independent environments. Within each layer, two closed-form fundamental solutions which satisfy the separated depth-dependent wave equation are formulated with complex analogues to be used in evanescent regions. Piecewise-continuous potential mode solutions are constructed by satisfying an upper boundary condition and extending through isovelocity layers to the bottom. The value of the Wronskian for the Green’s function thus obtained is used to locate the eigenvalues of the normal modes comprising the propagating field. A closed form solution in a modified two-layer Pekeris waveguide with step-wise changes in the upper layer sound speed with range is used to study horizontal mode coupling through a series of such environments and to predict propagation loss as a function of range.

Focusing phenomenon and near-trapped modes of SH waves

In this study, the null-field boundary integral equation method (BIEM) and the image method are used to solve the SH wave scattering problem containing semi-circular canyons and circular tunnels. To fully utilize the analytical property of circular geometry, the polar coordinates are used to expand the closed-form fundamental solution to the degenerate kernel, and the Fourier series is also introduced to represent the boundary density. By collocating boundary points to match boundary condition on the boundary, a linear algebraic system is constructed. The unknown coefficients in the algebraic system can be easily determined. In this way, a semi-analytical approach is developed. Following the experience of near-trapped modes in water wave problems of the full plane, the focusing phenomenon and near-trapped modes for the SH wave problem of the half-plane are solved, since the two problems obey the same mathematical model. In this study, it is found that the SH wave problem containing two semi-circular canyons and a circular tunnel has the near-trapped mode and the focusing phenomenon for a special incident angle and wavenumber. In this situation, the amplification factor for the amplitude of displacement is over 300.

Analysis of a nonlinear crack in a piezoelectric half-space via displacement discontinuity method

In this paper, we derive the exact closed-form fundamental solutions due to uniform extended displacement discontinuities over a triangular element in a piezoelectric half-space. Using the triangular elements to partition the penny-shaped crack, the triangular element fundamental solutions are verified by comparing with the existing analytical solution associated with the penny-shaped crack. The polarization saturation model is then applied to an elliptical crack in the piezoelectric half-space, and the resulting nonlinear fracture problem is solved by combing the triangular element fundamental solutions and the displacement discontinuity method. The electric yielding zone and the extended field intensity factors are obtained by an iterative approach. The effects of the applied mechanical load and electric displacement, the polarization saturation in the yielding zone, and the aspect ratio of the elliptical crack on the yielding zone size and field intensity factors are discussed through numerical examples.

Formulation of the MFS for the two-dimensional Laplace equation with an added constant and constraint

Motivated by the incompleteness of single-layer potential approach for the interior problem with a degenerate-scale domain and the exterior problem with bounded potential at infinity, we revisit the method of fundamental solutions (MFS). Although the MFS is an easy method to implement, it is not complete for solving not only the interior 2D problem in case of a degenerate scale but also the exterior problem with bounded potential at infinity for any scale. Following Fichera׳s idea for the boundary integral equation, we add a free constant and an extra constraint to the traditional MFS. The reason why the free constant and extra constraints are both required is clearly explained by using the degenerate kernel for the closed-form fundamental solution. Since the range of the single-layer integral operator lacks the constant term in the case of a degenerate scale for a two dimensional problem, we add a constant to provide a complete base. Due to the rank deficiency of the influence matrix in the case of a degenerate scale, we can promote the rank by simultaneously introducing a constant term and adding an extra constraint to enrich the MFS. For an exterior problem, the fundamental solution does not contain a constant field in the degenerate kernel expression. To satisfy the bounded potential at infinity, the sum of all source strengths must be zero. The formulation of the enriched MFS can solve not only the degenerate-scale problem for the interior problem but also the exterior problem with bounded potential at infinity. Finally, three examples, a circular domain, an infinite domain with two circular holes and an eccentric annulus were demonstrated to see the validity of the enriched MFS.

A semi-analytical method for near-trapped mode and fictitious frequencies of multiple scattering by an array of elliptical cylinders in water waves

In this paper, we focus on the water wave scattering by an array of four elliptical cylinders. The null-field boundary integral equation method (BIEM) is used in conjunction with degenerate kernels and eigenfunctions expansion. The closed-form fundamental solution is expressed in terms of the degenerate kernel containing the Mathieu and the modified Mathieu functions in the elliptical coordinates. Boundary densities are represented by using the eigenfunction expansion. To avoid using the addition theorem to translate the Mathieu functions, the present approach can solve the water wave problem containing multiple elliptical cylinders in a semi-analytical manner by introducing the adaptive observer system. Regarding water wave problems, the phenomena of numerical instability of fictitious frequencies may appear when the BIEM/boundary element method (BEM) is used. Besides, the near-trapped mode for an array of four identical elliptical cylinders is observed in a special layout. Both physical (near-trapped mode) and mathematical (fictitious frequency) resonances simultaneously appear in the present paper for a water wave problem by an array of four identical elliptical cylinders. Two regularization techniques, the combined Helmholtz interior integral equation formulation (CHIEF) method and the Burton and Miller approach, are adopted to alleviate the numerical resonance due to fictitious frequency.

Electrostatic potential of a point charge in a Brans-Dicke Reissner-Nordström field

We consider the Brans-Dicke Reissner-Nordstrom spacetime in isotropic coordinates and the electrostatic field of an electric point charge placed outside its surface of inversion. We treat the static electric point charge as a linear perturbation on the Brans-Dicke Reissner-Nordstrom background. We develop a method based upon the Copson method to convert the governing Maxwell equation on the electrostatic potential generated by the static electric point charge into a solvable linear second order ordinary differential equation. We obtain a closed form fundamental solution of the curved space Laplace equation arising from the background metric, which is shown to be regular everywhere except at the point charge and its image point inside the surface of inversion. We also develop a method that demonstrates that the solution does not contain any other charge that may creep into the region that lies beyond the surface of inversion and which is not covered by the isotropic coordinates. The Brans-Dicke Reissner-Nordstrom spacetime therefore is linearly stable under electrostatic perturbations. This stability result includes the three degenerate cases of the fundamental solution that correspond to the Brans Type 1, the Reissner-Nordstrom and the Schwarzschild background spacetimes.

A Semianalytical Approach for a Nonconfocal Suspended Strip in an Elliptical Waveguide

A problem of a nonconfocal suspended strip in an elliptical waveguide is analyzed by using a semianalytical approach, which is the so-called null-field boundary integral-equation method (BIEM). The null-field BIEM is proposed by introducing the idea of null field, degenerate kernels, and eigenfunction expansion to improve the conventional dual boundary-element method (BEM). A closed-form fundamental solution can be expressed in terms of the degenerate kernel containing the Mathieu and modified Mathieu functions in the elliptical coordinates. Boundary densities are represented by using the eigenfunction expansion. By this way, the efficiency is promoted in three aspects: analytical boundary integral without numerical error, natural bases for boundary densities, and exact description of boundary geometry. Due to the semianalytical formulation, the null-field BIEM can fully capture the property of geometry and the error only occurs from the truncation of the number of the eigenfunction expansion terms in the real implementation. The present method is also a kind of meshless method since only boundary nodes are needed to construct influence matrices instead of using boundary elements. Both TE and TM cases are considered in this paper. To verify the validity of the present method, the dual BEM and finite-element method are also utilized to provide cutoff wavenumbers. Besides, the analytical solution of a confocal elliptical waveguide can be derived by using the present method. After comparing with published data, good agreement is made.

Exact and complete fundamental solutions for penny-shaped crack in an infinite transversely isotropic thermoporoelastic medium: mode I problem

This paper examines the problem of a penny-shaped crack in a thermoporoelastic body. On the basis of the recently developed general solutions for thermoporoelasticity, appropriate potentials are suggested and the governing equations are solved in view of the similarity to those for pure elasticity. Exact and closed form fundamental solutions are expressed in terms of elementary functions. The singularity behavior is then discussed. The present solutions are compared with those in literature and an excellent agreement is achieved. Numerical calculations are performed to show the influence of the material parameters upon the distribution of the thermoporoelastic field. Due to its ideal property, the present solution is a natural benchmark to various numerical codes and simplified analyses.

Water wave problems using null-field boundary integral equations: ill-posedness and remedies

In this article, we focus on the hydrodynamic scattering of water wave problems containing circular and/or elliptical cylinders. Regarding water wave problems, the phenomena of numerical instability due to fictitious frequencies may appear when the boundary element method (BEM) is used. We examine the occurring mechanism of fictitious frequency in the BEM through a water wave problem containing an elliptical cylinder. In order to study the fictitious frequency analytically, the null-field boundary integral equation method in conjunction with degenerate kernels is employed to derive the analytical solution. The modal participation factor for the numerical instability of zero divided by zero can be exactly determined in a continuous system even though the circulant matrix cannot be obtained in a discrete system for the elliptical case. It is interesting to find that irregular values depend on the geometry of boundaries as well as integral representations and happen to be zeros of the mth-order (even or odd) modified Mathieu functions of the first kind or their derivatives. To avoid using the addition theorem to translate the Bessel functions to the Mathieu functions, the present approach can solve for the water wave problem containing circular and/or elliptical cylinders at the same time in a semi-analytical manner by using the adaptive observer system. The closed-form fundamental solution is expressed in terms of the degenerate kernel in the polar and elliptic coordinates for circular and elliptical cylinders, respectively. Three examples are considered to demonstrate the validity of the present approach, including an elliptical cylinder, two parallel identical elliptical cylinders and one circular cylinder and one elliptical cylinder. Finally, two regularization techniques, the combined Helmholtz interior integral equation formulation method and the Burton and Miller approach, are adopted to alleviate the numerical resonance due to fictitious frequency.

SH-wave scattering by a semi-elliptical hill using a null-field boundary integral equation method and a hybrid method

Following the success of seismic analysis of a semi-circular hill, the problem of SH-wave scattering by a semi-elliptical hill is revisited by using the null-field boundary integral equation method (BIEM). To fully use the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-elliptical hill scattering problem, the problem is decomposed to two regions to produce elliptical boundaries by using the technique of taking free body. One is the half-plane problem containing a semi-elliptical boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial elliptical boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by an elliptical boundary. The degenerate kernel in the elliptic coordinates for two subdomains is used to expand the closed-form fundamental solution. The semi-analytical formulation in companion with matching boundary conditions yields six constraint equations. Instead of finding admissible wave-expansion bases, our null-field BIEM in conjunction with degenerate kernels has the five features over the conventional BIEM/BEM, (1) free of calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless and (5) well-posed system. All numerical results are compared well with those of using the hybrid method which is also described in this paper. It is interesting to find that a focusing phenomenon is also observed in this study.

SH-wave diffraction by a semi-circular hill revisited: A null-field boundary integral equation method using degenerate kernels

Following the success of seismic analysis of a canyon [1], the problem of SH-wave diffraction by a semi-circular hill is revisited using the null-field boundary integral equation method (BIEM). To fully utilize the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-circular hill scattering problem, the problem is decomposed into two regions to produce circular boundaries using the technique of taking free body. One is the half-plane problem containing a semi-circular boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial full circular boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by a circular boundary. The degenerate kernel in the polar coordinates for two subdomains is utilized for the closed-form fundamental solution. The semi-analytical formulation along with matching boundary conditions yields six constraint equations. Instead of finding admissible wave expansion bases, our null-field BIEM approach in conjunction with degenerate kernels have five features over the conventional BIEM/BEM: (1) free from calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless and (5) well-posed system. All the numerical results are comparing well with the available results in the literature. It is interesting to find that a focusing phenomenon is also observed in this study.

On modelling SH‐waves in a class of inhomogeneous anisotropic media via the Boundary Element Method

AbstractWe present here a Boundary Element Method (BEM) implementation for SH harmonic waves in a class of inhomogeneous anisotropic media. The inhomogeneity is assumed to be the same not only for the stiffnesses, but also for the density. The implementation is based on a closed form fundamental solution for SH waves derived by Daros [C. H. Daros, A fundamental solution for SH‐waves in class of inhomogeneous anisotropic media, Int. J. Eng. Science 46 (2008) 809–817]. We show numerical results obtained by the traditional boundary integral equation. Moreover, the non‐hypersingular traction based BEM is also implemented, allowing the modelling of cracks in inhomogeneous anisotropic media. We obtain numerical results for the stress intensity factors (SIF) which are compared to previous published results.

In-plane vibrational analysis of rotating curved beam with elastically restrained root

This study analyzes the in-plane free vibration of a rotating curved beam with an elastically restrained root. Neglecting the effects of shear deformation and the Coriolis force, governing differential equations are derived for the coupled bending–extensional vibration of the curved beam using Hamilton's principle and a consistent linearization approach. Explicit relations are constructed to describe the correlation between the axial and radial displacements of the beam. These relations are then used to transform the coupled governing differential equations into a sixth-order ordinary differential equation expressed in terms of the radial displacement variable only. An exact closed-form fundamental solution of the transformed system is then derived. Finally, the respective effects of the arc angle, the rotational speed, the hub radius and the root spring constants on the natural frequencies and divergent instability characteristics of a curved rotating beam are systematically examined and compared with those observed for a straight cantilever beam.

The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids

This paper describes the method of fundamental solutions (MFS) to solve three-dimensional elastostatic problems of transversely isotropic solids. The desired solution is represented by a series of closed-form fundamental solutions, which are the displacement fields due to concentrated point forces acting on the transversely isotropic material. To obtain the unknown intensities of the fundamental solutions, the source points are properly located outside the computational domain and the boundary conditions are then collocated. Furthermore, the closed-form traction fields corresponding to the previously published point force solutions are reviewed and addressed explicitly in suitable forms for numerical implementations. Three numerical experiments including Dirichlet, Robin, and peanut-shaped-domain problems are carried out to validate the proposed method. It is found that the method performs well for all the three cases. Furthermore, a rescaling method is introduced to improve the accuracy of Robin problem with noisy boundary data. In the spirits of MFS, the present meshless method is free from numerical integrations as well as singularities.