Boundary integral equation formulation for the generalized micro-polar thermoviscoelasticity

A formulation of the boundary integral equation method for generalized linear micro-polar thermoviscoelasticity is given. Fundamental solutions, in Laplace transform domain, of the corresponding differential equations are obtained. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation. The results are applicable to the generalized thermoelasticity theories: Lord-Shulman with one relaxation time, Green-Lindsay with two relaxation times, Green-Naghdi theories, and Chandrasekharaiah and Tzou with dual-phase lag, as well as to the dynamic coupled theory. The cases of generalized linear micro-polar thermoviscoelasticity of Kelvin-Voigt model, generalized linear thermoviscoelasticity and generalized thermoelasticity can be obtained from the given results.     

A new integral equation formulation for the analysis of crack-inclusion interactions

A new integral equation method for the analysis of the interactions between cracks and elastic inclusions embedded in a two-dimensional, linearly elastic, isotropic infinite medium subjected to in-plane force is presented. By distributing dislocations along the crack lines and forces along the matrix-inclusion interfaces, a set of coupled integral equations is obtained. The discretization procedure of the integrals involved is discussed and the relations between the stress intensity factors and the values of the dislocation functions at the respective crack tips are derived. Several sample problems are presented in order to determine the versatility and the accuracy of this approach.     

On the boundary integral formulation of thermo-viscoelasticity theory

A formulation of the boundary integral equation method for generalized thermo-viscoelasticity is given. Fundamental solutions of the corresponding differential equations are obtained. An outline of the implementation of the boundary element method is discussed for the solution of the above boundary equations. Special emphasis is given to the representation of the primary fields, namely, temperature and displacement.     

Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics

The field equations of three-dimensional elastostatics are transformed to boundary integral equations. The elastic body is divided into subregions, and the surface and interfaces are represented by quadrilateral and triangular elements with quadratic variation of geometry and linear, quadratic or cubic variation of displacement and traction with respect to intrinsic co-ordinates. The integral equation is discretized for each subregion, and a system of banded form obtained. For the integration of kernel-shape function products, Gaussian quadrature formulae are chosen according to upper bounds for error in terms of derivatives of the integrands. Use of the integral formulation is illustrated by the analysis of a prestressed concrete nuclear reactor pressure vessel.     

Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites

A novel computational model is presented using the eigenstrain formulation of the boundary integral equations for modeling the particle-reinforced composites. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix. The eigenstrains of inhomogeneity are determined with the aid of the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. The solution scale of the inhomogeneity problem with the present model is greatly reduced since the unknowns appear only on the boundary of the solution domain. The overall elastic properties are solved using the newly developed boundary point method for particle-reinforced inhomogeneous materials over a representative volume element with the present model. The effects of a variety of factors related to inhomogeneities on the overall properties of composites as well as on the convergence behaviors of the algorithm are studied numerically including the properties and shapes and orientations and distributions and the total number of particles, showing the validity and the effectiveness of the proposed computational model.     

Boundary element method based on a new second kind integral equation formulation

A boundary integral equation (BIE) formulation for elasticity problems with mixed boundary conditions, proposed by Parton and Perlin (Mathematical Methods of the Theory of Elasticity, Mir, Moscow, 1984), is implemented in this paper using quadratic boundary elements. The formulation is specialised to Stokes flow problems by setting the Poisson ratio to 0·5 in the relevant kernels. The implementation is used to analyse non-trivial three dimensional problems in elasticity and Stokes flows. The results compare well with those obtained by a direct boundary element method. An outline of the extension of the formulation to non-linear problems is also given.     

An integral formulation applied to the diffusion and Boussinesq equations

A new integral method is proposed here to solve the diffusion equation (confined flow) and the Boussinesq equation (unconfined flow) in a two-dimensional porous medium. The method, based on Green's theorem, derives its integral representation from the portion of the original differential equation with the highest space derivatives so that the resulting kernel of the integral representation is not time dependent. Compared to an earlier integral formulation, namely the direct Green function, based on the same theorem, the kernel is simpler so that the present theory provides a more efficient numerical model without compromising accuracy. An iterative scheme is employed along with the theory to achieve solutions to the non-linear Boussinesq equation. Concepts used in the finite difference and finite element methods enable simplification of the temporal derivative. The method is tested with success on a number of numerical examples from groundwater flow.     

A classical iterative theory based on the Langevin equation for two-dimensional nonlinear terahertz spectroscopy

A classical iterative theory based on the Langevin equation is presented to obtain the nonlinear response of a system and simulate two-dimensional (2D) nonlinear terahertz (THz) spectroscopy (2DTS). Compared with the widely used method of calculating the multi-time correlation function or the Poison brackets, we start from the classical Langevin equation and use an iterative method to obtain any order of nonlinear response. The anharmonic potential (AHP) and nonlinear coordinates dependence of the dipole moment (NDM) are two types of nonlinear sources introduced here. Results are derived for general three-pulse processes with nonlinear sources, AHP or NDM, separately and with the combination of both. Only the simulative 2DTS results for the single mode case with impulsive incident THz fields are presented.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

A finite element formulation based on Nadai's deformation theory for elasto-plastic analysis

A numerical method of the Newton-Raphson type is presented for elasto-plastic analysis using the finite element method. The method is developed from Nadai's deformation theory and Hooke's law. Numerical examples are used to show that the method provides very rapid solution convergence.     

A fast spectral Galerkin method for hypersingular boundary integral equations in potential theory

This research is focused on the development of a fast spectral method to accelerate the solution of three-dimensional hypersingular boundary integral equations of potential theory. Based on a Galerkin approximation, the fast Fourier transform and local interpolation operators, the proposed method is a generalization of the precorrected-FFT technique to deal with double-layer potential kernels, hypersingular kernels and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are included to illustrate the performance of the method. The US Government retains a nonexclusive royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for US Government purposes.     

A numerical solution for integral equations

A method for the numerical approximation for the solution of Fredholm integral equations of the second kind is presented. The approximation is obtained using cubic B-splines as co-ordinate functions in the variational formulation of the problem.     

A new SMA shell element based on the corotational formulation

Aim of this paper is to develop a new shape memory alloy (SMA) facet-shell finite element accounting for material and geometric nonlinearities. A corotational formulation is exploited, able to filter out large rigid-body motions from the element transformation. Accordingly, a geometrically linear core-element is employed, along with a SMA constitutive model formulated in the small strain framework. In particular, in accordance with the formulation of the classical thin shell theory, a plane-stress SMA model accounting for the pseudo-elastic as well as the shape memory effect is adopted. The time integration of the evolutive equation is performed developing a step-by-step backward-Euler numerical procedure. A highly efficient implementation of the corotational machinery is used, endowed with a fully consistent tangent stiffness. Applications are carried out for assessing the performances of the developed computational procedure and to investigate on some interesting engineering examples. The numerical results show the effectiveness of the proposed shell element, whose simplicity makes it attractive for the design of new advanced SMA-based devices undergoing significant configuration changes during their operation.     

A nonlinear thick plate formulation based on the modified strain gradient theory

A geometric nonlinear first-order shear deformation theory-based formulation is presented to analyze microplates. The formulations derived herein are based on a modified strain gradient theory and the von Karman nonlinear strains. The modified strain gradient theory includes five material length scale parameters capable to capture the size effects in small scales. The governing equations of motion and the most general form of boundary conditions of an arbitrary-shaped plate are derived using the principle of virtual displacements. The analysis is general and can be reduced to the modified couple stress plate model or the classical plate model.     

A nonlinear Timoshenko beam formulation based on the modified couple stress theory

This paper presents a nonlinear size-dependent Timoshenko beam model based on the modified couple stress theory, a non-classical continuum theory capable of capturing the size effects. The nonlinear behavior of the new model is due to the present of induced mid-plane stretching, a prevalent phenomenon in beams with two immovable supports. The Hamilton principle is employed to determine the governing partial differential equations as well as the boundary conditions. A hinged–hinged beam is chosen as an example to delineate the nonlinear size-dependent static and free-vibration behaviors of the derived formulation. The solution for the static bending is obtained numerically. The solution for the free-vibration is presented analytically utilizing the method of multiple scales, one of the perturbation techniques.     

Multinode finite element based on boundary integral equations

A finite element constructed on the basis of boundary integral equations is proposed. This element has a flexible shape and arbitrary number of nodes. It also has good approximation properties. A procedure of constructing an element stiffness matrix is demonstrated first for one-dimensional case and then for two-dimensional steady-state heat conduction problem. Numerical examples demonstrate applicability and advantages of the method. © 1998 John Wiley & Sons, Ltd.     

A new method of solving the integral equations of radiation heat transfer

We present a method for calculating the radiation heat transfer between bodies and a new approach to the solution of the corresponding integral equations. We show, in fact, that the first approximation describes the solution of these equations with sufficiently high accuracy and is, in some cases, exact.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 51, No. 6, pp. 1031–1040, December, 1986.