Volume Integral Equation Method Solution for Spheroidal Inclusion Problem

In this paper, the volume integral equation method (VIEM) is introduced for the numerical analysis of an infinite isotropic solid containing a variety of single isotropic/anisotropic spheroidal inclusions. In order to introduce the VIEM as a versatile numerical method for the three-dimensional elastostatic inclusion problem, VIEM results are first presented for a range of single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix under uniform remote tensile loading. We next considered single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix under remote shear loading. The authors hope that the results using the VIEM cited in this paper will be established as reference values for verifying the results of similar research using other analytical and numerical methods.

Algorithm for inversion of resistivity logging-while-drilling data in 2D pixel-based model

Multi-frequency and multi-component extra-deep azimuthal resistivity measurements with depth of investigation of a few tens of meters provide advanced possibilities for mapping of complex reservoir structures. Inversion of the induction measurements set becomes an important technical problem. We present a regularized Levenberg–Marquardt algorithm for inversion of resistivity measurements in a 2D environment model with pixel-based resistivity distribution. The cornerstone of the approach is an efficient parallel algorithm for computation of resistivity tool signals and its derivatives with respect to the pixel conductivities using volume integral equation method. Numerical tests of the suggested approach demonstrate its feasibility for near real time inversion.

Volume Integral Equation Method (VIEM)

A number of analytical techniques are available for the stress analysis of inclusion problems when the geometries of inclusions are simple (e.g., cylindrical, spherical or ellipsoidal) and when they are well separated [9, 41, 52]. However, these approaches cannot be applied to more general problems where the inclusions are anisotropic and arbitrary in shape, particularly when their concentration is high. Thus, stress analysis of heterogeneous solids or analysis of elastic wave scattering problems in heterogeneous solids often requires the use of numerical techniques based on either the finite element method (FEM) or the boundary integral equation method (BIEM). However, these methods become problematic when dealing with elastostatic problems or elastic wave scattering problems in unbounded media containing anisotropic and/or heterogeneous inclusions of arbitrary shapes. It has been demonstrated that the volume integral equation method (VIEM) can overcome such difficulties in solving a large class of inclusion problems [6,10,20,21,28–30]. One advantage of the VIEM over the BIEM is that it does not require the use of Green’s functions for anisotropic inclusions. Since the elastodynamic Green’s functions for anisotropic media are extremely difficult to calculate, the VIEM offers a clear advantage over the BIEM. In addition, the VIEM is not sensitive to the geometry or concentration of the inclusions. Moreover, in contrast to the finite element method, where the full domain needs to be discretized, the VIEM requires discretization of the inclusions only.

Three-Dimensional Volume Integral Equation Method for Solving Isotropic/Anisotropic Inhomogeneity Problems

In this paper, the volume integral equation method (VIEM) is introduced for the analysis of an unbounded isotropic solid composed of multiple isotropic/anisotropic inhomogeneities. A comprehensive examination of a three-dimensional elastostatic VIEM is introduced for the analysis of an unbounded isotropic solid composed of isotropic/anisotropic inhomogeneity of arbitrary shape. The authors hope that the volume integral equation method can be used to compute critical values of practical interest in realistic models of composites composed of strong anisotropic and/or heterogeneous inhomogeneities of arbitrary shapes.

The volume integral equation method in magnetostatic problem

This article addresses the issues of volume integral equation method application to magnetic system calculations. The main advantage of this approach is that in this case finding the solution of equations is reduced to the area filled with ferromagnetic. The difficulty of applying the method is connected with kernel singularity of integral equations. For this reason in collocation method only piecewise constant approximation of unknown variables is used within the limits of fragmentation elements inside the famous package GFUN3D. As an alternative approach the points of observation can be replaced by integration over fragmentation element, which allows to use approximation of unknown variables of a higher order.In the presented work the main aspects of applying this approach to magnetic systems modelling are discussed on the example of linear approximation of unknown variables: discretisation of initial equations, decomposition of the calculation area to elements, calculation of discretised system matrix elements, solving the resulting nonlinear equation system. In the framework of finite element method the calculation area is divided into a set of tetrahedrons. At the beginning the initial area is approximated by a combination of macro-blocks with a previously constructed two-dimensional mesh at their borders. After that for each macro-block separately the procedure of tetrahedron mesh construction is performed. While calculating matrix elements sixfold integrals over two tetrahedra are reduced to a combination of fourfold integrals over triangles, which are calculated using cubature formulas. Reduction of singular integrals to the combination of the regular integrals is proposed with the methods based on the concept of homogeneous functions. Simple iteration methods are used to solve non-linear discretized systems, allowing to avoid reversing large-scale matrixes. The results of the modelling are compared with the calculations obtained using other methods.

Parallel Volume Integral Equation Method for Two-Dimensional Elastodynamic Analysis

The parallel volume integral equation method (PVIEM) is applied for the analysis of two-dimensional elastic wave scattering problems in an unbounded isotropic solid containing various types of multiple multilayered anisotropic inclusions. It should be noted that the volume integral equation method (VIEM) does not require the use of the Green’s function for the anisotropic inclusion — only the Green’s function for the unbounded isotropic matrix is needed. A detailed analysis of the SH wave scattering problem is presented for various types of multiple multilayered orthotropic inclusions. Numerical results are presented for the elastic fields at the interfaces for square and hexagonal packing arrays of various types of multilayered orthotropic inclusions in a broad frequency range of practical interest. Standard parallel programming was used to speed up computation in the VIEM. The PVIEM enables us to investigate the effects of single/multiple scattering, fiber packing type, fiber volume fraction, single/multiple layer(s), multilayer’s shapes and geometry, isotropy/anisotropy, and softness/hardness of various types of multiple multilayered anisotropic inclusions on displacements and stresses at the interfaces of the inclusions and far-field scattering patterns. Also, powerful capabilities of the PVIEM for the analysis of general two-dimensional multiple scattering problems are investigated.