Isotropic Problems Research Articles

Symplectic Analytical Solutions for the Magnetoelectroelastic Solids Plane Problem in Rectangular Domain

The transversely isotropic magnetoelectroelastic solids plane problem in rectangular domain is derived to Hamiltonian system. In symplectic geometry space with the origin variables—displacements, electric potential, and magnetic potential, as well as their duality variables—lengthways stress, electric displacement, and magnetic induction, on the basis of the obtained eigensolutions of zero‐eigenvalue, the eigensolutions of nonzero‐eigenvalues are also obtained. The former are the basic solutions of Saint‐Venant problem, and the latter are the solutions which have the local effect, decay drastically with respect to distance, and are covered in the Saint‐Venant principle. So the complete solution of the problem is given out by the symplectic eigensolutions expansion. Finally, a few examples are selected and their analytical solutions are presented.

Analytical solution for a concentrated force on the free surface of a coated material

Based on the general solution of the displacement method for isotropic plane problems, the analytical solution for the plane problem of coated materials subjected to an arbitrary concentrated force on the free surface has been derived explicitly by using the image point method. The displacement functions are assumed to be the infinite series of the harmonic functions defined in the local coordinate systems with their origins placed at different image points. The harmonic functions corresponding to the higher-order image points can be deduced from those to the lower-order points by the recurrence formulae presented in this paper, and the first two harmonic functions are the displacement functions for the solution of a semi-infinite plane subjected to a concentrated force on the free surface. The theoretical formulae have been confirmed by numerical, finite-element-based, results in a special coated material.

Volume integral equation method for multiple isotropic inclusion problems in an infinite solid under tension or in-plane shear

A volume integral equation method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic inclusions subject to uniform remote tension or in-plane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix, and the central inclusion is carried out for square and hexagonal packing of the inclusions. The effects of the number of isotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central inclusion are also investigated in detail. The accuracy and efficiency of the method are examined in comparison with results obtained from analytical and finite element methods.

Formulation of the flow problem in anisotropic porous media using different coordinate transformations

To solve an anisotropic flow problem, we transform the problem into an equivalent isotropic problem. We expect that an anisotropic solution can be readily obtained from the equivalent isotropic solution. However, sometimes it is not the case. In this paper we use the drawdown equation of a fully-penetrated vertical well test, and the drawdown and buildup equations for a probe test to discuss when we can and when we cannot directly obtain an anisotropic solution from an isotropic solution in this paper. Due to the fact that transformation causes the changes in flow geometry and the shape of the wellbore or probe, we have to modify the equivalent isotropic solution to obtain the anisotropic solution. Finally, we present a case that the horizontal well productivity was incorrectly derived in a published paper, due to the misuse of the concept of anisotropy.

Seismic response of an embedded pile in a transversely isotropic half-space under incident P-wave excitations

A rigorous mathematical formulation is presented for the analysis of a thin cylindrical shell embedded in a transversely isotropic half-space under vertically incident P-wave excitation. By virtue of a set of ring-loads Green's functions for the shell and a group of dynamic fundamental solutions for the half-space under arbitrary interfacial dynamic loads, the problem is shown to be reducible to a pair of Fredholm integral equations. By utilizing an adaptive-gradient family capable of capturing regular-to-singular solution transitions smoothly, an accurate numerical procedure is developed. To assess the effect of material anisotropy on the dynamic load-transfer process, a set of comprehensive numerical results presented for various material and geometrical conditions. The accuracy of the proposed numerical scheme is confirmed by its comparison with a benchmark solution for the corresponding isotropic problem.

Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems

We investigate two algorithms involving the relaxation of either the given Dirichlet data (boundary temperatures) or the prescribed Neumann data (normal heat fluxes) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [26] applied to two-dimensional steady-state heat conduction Cauchy problems, i.e. Cauchy problems for the Laplace equation. The two mixed, well-posed and direct problems corresponding to each iteration of the numerical procedure are solved using a meshless method, namely the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the Laplace operator in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.

Performance Analysis of the Mesh‐Free Random Grid Method for Full‐Field Synthetic Strain Measurements

Abstract: In responding to the needs of the material characterization community, the recently developed mesh‐free random grid method (MFRGM) has been exhibiting very promising characteristics of accuracy, adaptability, implementation flexibility and efficiency. To address the design specification of the method according to an intended application, we are presenting a sensitivity analysis that aids into determining the effects of the experimental and computational parameters characterizing the MFRGM in terms of its performance. The performance characteristics of the MFRGM are mainly its accuracy, sensitivity, smoothing properties and efficiency. In this paper, we are presenting a classification of a set of parameters associated with the characteristics of the experimental set‐up and the random grid applied on the specimen under measurement. The applied sensitivity analysis is based on synthetic images produced from analytic solutions of specific isotropic and orthotropic elasticity boundary value problems. This analysis establishes the trends in the performance characteristics of the MFRGM that will enable the selection of the user controlled variables for a desired performance specification.

Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties

This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method.

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.

Discussion of Permeability Anisotropy Effect in Transformation

Summary To solve an anisotropic flow problem, the problem is transformed into an equivalent isotropic problem. It is expected that an anisotropic solution can be readily obtained from the equivalent isotropic solution. However, sometimes that is not the case. Using the drawdown equation of a fully penetrated vertical well test and the drawdown and buildup equations for a probe test, this paper discusses when it is both possible and impossible to directly obtain an anisotropic solution from an isotropic solution. Because of the fact that transformation causes the changes in flow geometry and the shape of the wellbore or probe, the equivalent isotropic solution must be modified to obtain the anisotropic solution. Finally, a case of how horizontal well productivity was incorrectly derived in a published paper --caused by the misuse of the concept of anisotropy--is presented. The full manuscript of SPE 114504, Discussion of Permeability Anisotropy Effect, is available as a supplement to this article.

The degeneration of image singularities from anisotropic materials to isotropic materials for an elastic half-plane

The degeneration of image singularities from an anisotropic material to an isotropic material for a half-plane is discussed in this study. The Green’s functions for anisotropic and isotropic half-planes with traction free boundary subjected to concentrated forces and dislocations have been obtained by many authors. It was commonly accepted that the solution of isotropic problem cannot be derived from anisotropic solutions. However, we believe that this possibility exists as we will demonstrate in this paper. Anisotropic materials include only image singularities of order O(1/ r) (i.e., forces and dislocations) existing on image points. There are many image points for anisotropic materials and the locations of these image points depend on the material constants. However, isotropic materials have only one image point with higher order image singularities (O(1/ r 2), O(1/ r 3)). From the analysis provided in this study, it is found that the higher order image singularities for an isotropic half-plane are generated by combining the concentrated forces and dislocations when an anisotropic material degenerates to an isotropic material. The solutions of higher order image singularities for isotropic material are dependent. Therefore, these image singularities can be combined to form only three or four simpler image singularities acting on an image point of the isotropic material.

A Cylindrically Anisotropic Tube Containing a Mixed Dislocation

AbstractThis study presents an analytical methodology for solving an elastic problem of a cylindrically anisotropic tube infused with eigenstrains. The general solutions for the particular case of a tube containing a mixed dislocation are also provided. The mainframe of this analysis is based on the state space formulation in conjunction with the theory of eigenstrain. By using the technique of Fourier series expansion and the theory of matrix algebra on solving the state equation, the expressions of solutions are not only explicit but also compact. The dislocation considered in this study is a mixed dislocation which can be viewed as a combination of edge dislocations and a screw dislocation. In order to strengthen the feasibility of this analysis, the strategy to determine the inverse of a singular matrix is thoroughly discussed, such that the general solutions can be smoothly applied to an isotropic tube problem. The results for an isotropic tube, which are reduced from the general forms, are compared with the well-established researches of related cases in the literature. The acceptable correspondences indicate the applicability of this study. An elastic problem of a cylindrically orthotropic tube containing a dislocation is also investigated as a demonstrating example. On this example, several particular phenomenons of stress distribution on the tube surface are presented in figures and discussed.

Long Time Behavior of a Two-Phase Optimal Design for the Heat Equation

We consider a two-phase isotropic optimal design problem within the context of the transient heat equation. The objective is to minimize the average of the dissipated thermal energy during a fixed time interval $[0,T]$. The time-independent material properties are taken as design variables. A full relaxation for this problem was established in [A. Münch, P. Pedregal, and F. Periago, J. Math. Pures Appl. (9), 89 (2008), pp. 225–247] by using the homogenization method. In this paper, we study the asymptotic behavior as T goes to infinity of the solutions of the relaxed problem and prove that they converge to an optimal relaxed design of the corresponding two-phase optimization problem for the stationary heat equation. Next we study necessary optimality conditions for the relaxed optimization problem under the transient heat equation and use those to characterize the microstructure of the optimal designs, which appears in the form of a sequential laminate of rank at most N, the spatial dimension. An asymptotic analysis of the optimality conditions lets us prove that, for T large enough, the order of lamination is, in fact, of at most $N-1$. Several numerical experiments in two dimensions complete our study.

A 3D shell-like approach using a natural neighbour meshless method: Isotropic and orthotropic thin structures

A three-dimensional shell-like approach for the analysis of composite thin plates and shells using a meshless method , the natural neighbour radial point interpolation method (NNRPIM), is presented. In the NNRPIM the nodal connectivity is enforced using the natural neighbour concept. The node-depending background mesh used in the numerical integration of the NNRPIM interpolation functions is entirely created from the unstructured nodal arrangement. The radial point interpolators are used to construct the NNRPIM interpolation functions, which possesses the delta Kronecker property, used in the Galerkin weak form. The novelty of this work lays on the development of a unique NNRPIM approach when 3D thin structures are considered. This new approach leads to remarkable results and it is extremely suitable to the composite structure problem. In order to demonstrate the effectiveness of the method the 3D shell-like NNRPIM analysis is used to solve several isotropic and orthotropic thin plates and shells problems.

Some simple Cartesian solutions to plane non-homogeneous elasticity problems

Abstract The paper presents some simple solutions to plane stress non-homogeneous isotropic elasticity problems described in Cartesian coordinates. The general problem is formulated in terms of the usual Airy stress function allowing spatial variation in Young’s modulus while keeping Poisson’s ratio constant. The resulting general equation is lengthy and involves various derivatives of the stress function and first and second order derivatives of the modulus distribution. Two inverse schemes are presented which greatly simplify the general governing equation and allow simple exact solutions to be generated. The first scheme employs simple biharmonic polynomial forms for the stress function thereby automatically giving stress fields identical to the homogeneous case. The governing equation is reduced to a form involving only modulus gradation terms and can often be easily solved for the allowable modulus distribution. The resulting displacement fields are then determined by standard methods. A second related method uses special simplifying elastic modulus variation to greatly reduce the general equation thereby allowing simple integration to determine the solution.

Proof of the Strong Eshelby Conjecture for Plane and Anti-plane Anisotropic Inclusion Problems

Based on the Stroh formalism for anisotropic elasticity and the complex variable function method, we prove in this paper that the strong Eshelby conjecture holds for simply-connected anisotropic inclusion problems under plane or anti-plane deformation. The interfaces can be either perfect or dislocation-like. For these inclusion problems, if the induced stress field inside the inclusion is uniform for a single uniform eigenstrain, the inclusion is of the elliptic shape. Thanks to the generality of the proof method, we obtain also alternative proofs of the strong Eshelby conjecture for isotropic inclusion problems, which are given in the Appendix.

A Compact Procedure for Discretization of the Anisotropic Diffusion Operator

This article presents a conservative finite-volume-based scheme for discretization of the anisotropic diffusion operator, which is simple to implement and numerically conservative. The procedure is an extension of a standard method used for discretization of the isotropic diffusion operator in finite-volume methods. Consequently, the technique can be easily implemented in codes originally developed to solve isotropic diffusion problems. The method is applied in an unstructured finite-volume solver and assessed by solving the following four problems: (1) steady conduction in an anisotropic block, (2) anisotropic conduction with a source term, (3) anisotropic conduction in a hollow block, and (4) heterogeneous anisotropic conduction with embedded point heat sources. Calculations are performed for a wide range of parameters using triangular/quadrilateral cells on grid systems with sizes ranging between 102 and 3 × 105 control volumes, and convergence is accelerated through the use of an algebraic multigrid method. Converged solutions are obtained for all problems and on all grid systems used, and the results are in excellent agreement with the exact analytical solutions.

The method of fundamental solutions with dual reciprocity for three‐dimensional thermoelasticity under arbitrary body forces

PurposeThe purpose of this paper is to develop a meshless numerical method for three‐dimensional isotropic thermoelastic problems with arbitrary body forces.Design/methodology/approachThis paper combines the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method (MFS‐DRM) to solve three‐dimensional isotropic thermoelastic problems with arbitrary body forces. In the DRM, the arbitrarily distributed temperature and body force are approximated by polyharmonic splines with augmented polynomial basis, whose particular solutions and the corresponding tractions are reviewed and given explicitly. The MFS is then applied to solve the complementary solution. Numerical experiments of Dirchlet, Robin, and peanut‐shaped‐domain problems are carried out to validate the method.FindingsIn literature, it is commented that the Gaussian elimination can be used reliably to solve the MFS equations for non‐noisy boundary conditions. For noisy boundary conditions, the truncated singular value decomposition (TSVD) is more accurate than the Gaussian elimination. In this paper, it was found that the particular solutions obtained by the DRM act like noises and the use of TSVD improves the accuracy.Originality/valueIt is the first time that the MFS‐DRM is derived to solve three‐dimensional isotropic thermoelastic problems with arbitrary body forces.

Curvilinear Coordinate Condition for Antiplane Strain Deformation in Power Law Plastic Solid

A desirable curvilinear coordinate system for problems in antiplane strain is one in which the shear stress acting across orthogonal trajectories is either zero or has the maximum value. In elastic isotropic solid problems any coordinate system given by z ≡ x+iy = f ( w ) ≡ f (u + iv) satisfies this condition. This paper examines some requirements for curvilinear coordinate systems with a power law plastic solid (of power m). In particular, if the curvilinear trajectories are given by x = x (u, v) and y = y (u, v) the trajectories must satisfy the equation where 7 represents either x or y.

Efficient Simulation of Elastic Waves Propagation in Anisotropic Media

AbstractThe paper deals with finite–difference (f‐d) approach to simulation of elastic waves' propagation in anisotropic elastic media with general symmetry. Any implementation of this approach claims resolution of two key problems: construction of an effective f‐d scheme itself; we propose to use the Lebedev's scheme (LS) being a natural generalization of Virieux staggered grid scheme (VS) widely used for isotropy; we prove that LS possesses better dispersion properties in comparison with well known Rotated Staggered Grids Scheme (RSGS). stable domain distension. The Perfectly Matched Layers(PML) useful for isotropic problems can be unstable in the case of anisotropy. Lebedev scheme allows one to use Optimal Grids (OG) which gives a possibility to implement efficient and low cost domain distension. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)