Comparative Analysis of 3D Viscoelastic and Poroelastic Dynamic Boundary-Element Modelling

2014 ◽  
Vol 709 ◽  
pp. 186-189
Author(s):  
L.A. Igumnov ◽  
A.A. Ipatov ◽  
E.A. Lebedeva
Keyword(s):  
Boundary Value Problems ◽  
Boundary Element ◽  
Integral Transform ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Three Dimensional ◽  
Maxwell Model ◽  
Boundary Integral ◽  
Weakly Singular Kernel ◽  
Dynamic Boundary

3D dynamic boundary-value problems of linear viscoelasticity and poroelasticity are considered. Laplace integral transform and its numerical inversion are used. Classical viscoelastic models, such as Maxwell model, KelvinVoigt model, standard linear solid model, and model with weakly singular kernel (Abel type) are considered. Boundary integral equations (BIE) method is developed to solve three-dimensional boundary-value problems. A numerical modelling of wave propagation is done by means of boundary element approach.

On mixed boundary value problems for the Helmholtz equation

1977 ◽  
Vol 77 (1-2) ◽  
pp. 65-77 ◽  
Author(s):  
R. Kress ◽  
G. F. Roach
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Theoretic Approach ◽  
Mixed Boundary Value Problems

SynopsisExistence and uniqueness theorems are obtained for a class of mixed boundary value problems associated with the three-dimensional Helmholtz equation. In this context the boundary of the region of interest is assumed to consist of the union of a finite number of disjoint, closed, bounded Lyapunov surfaces on some of which are imposed Dirichlet conditions whilst Neumann conditions are imposed on the remainder. An integral equation method is adopted throughout. The required boundary integral equations are generated by a modified layer theoretic approach which extends the work of Brakhage and Werner [1] and Leis [2, 3].

Boundary Element Formulation for Numerical Surface Wave Modelling in Poroviscoelastisity

2016 ◽  
Vol 685 ◽  
pp. 172-176 ◽  
Author(s):  
Leonid Igumnov ◽  
S.Yu. Litvinchuk ◽  
A.A. Belov ◽  
A.A. Ipatov
Keyword(s):  
Boundary Element ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Dynamic Responses ◽  
Element Formulation ◽  
Weakly Singular Kernel ◽  
Viscoelastic Parameters ◽  
Viscoelastic Effects ◽  
Biot’S Equations

Problems of the poroviscodynamics are considered. Theory of poroviscoelasticity is based on Biot’s equations of fluid saturated porous media under assumption that the skeleton is viscoelastic. Viscoelastic effects of solid skeleton are modeled by mean of elastic-viscoelastic correspondence principle, using such viscoelastic models as a standard linear solid model and model with weakly singular kernel. The fluid is taken as original in Biot’s formulation without viscoelastic effects. Boundary integral equations method is applied to solve three-dimensional boundary-value problems. Boundary-element method with mixed discretization and matched approximation of boundary functions is used. Solution is obtained in Laplace domain, and then Durbin’s algorithm of numerical inversion of Laplace transform is applied to perform solution in time domain. An influence of viscoelastic parameters on dynamic responses is studied. Numerical example of the surface waves modelling is considered.

Boundary Element Solution for Dynamics of Inhomogeneous Poroviscoelastic Beam

2018 ◽  
Vol 769 ◽  
pp. 352-357
Author(s):  
Leonid A. Igumnov ◽  
Aleksandr A. Ipatov
Keyword(s):  
Boundary Element ◽  
Pore Pressure ◽  
Numerical Solutions ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Dynamic Responses ◽  
Numerical Inversion ◽  
Element Approximation

The present paper is dedicated to numerical solution of boundary-value problems for piecewise homogeneous solids in terms of linear three-dimensional poroviscoelasticity. Mathematical model of poroviscoelastic material is based on Biot's model of poroelasticity. Viscoelastic effects refer to a skeleton of porous material and are described through the correspondence principle. Standard linear solid model is employed. In order to study the boundary-value problem boundary integral equations method is applied, and to find their solutions boundary element method for obtaining numerical solutions. A direct version of the BIE method is developed. The boundary-element scheme is constructed using: regularized BIE’s, a matched element-by-element approximation, adaptive numerical integration in combination with a singularity-reducing algorithm. The solution of the original problem is constructed in Laplace transforms, with the subsequent application of the algorithm for numerical inversion. Modified Durbin’s algorithm of numerical inversion of Laplace transform is applied to perform solution in time domain. The problem of poroviscoelastic prismatic solid clamped from one end and free at another is considered. The solid is composed of two subdomains. Heaviside-type load is applied to a free end of the solid. Viscosity parameter influence on dynamic responses of displacements, pore pressure and tractions is studied. Numerical results for displacements and pore pressure when subdomains are modelled with different viscoelastic properties are presented.

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

Boundary Element Method Analysis of Three-Dimensional Thermoelastic Fracture Problems Using the Energy Domain Integral

2005 ◽  
Vol 73 (6) ◽  
pp. 959-969 ◽  
Author(s):  
R. Balderrama ◽  
A. P. Cisilino ◽  
M. Martinez
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Auxiliary Function ◽  
Boundary Integral ◽  
Crack Advance ◽  
Domain Integral ◽  
Energy Domain ◽  
Element Method

A boundary element method (BEM) implementation of the energy domain integral (EDI) methodology for the numerical analysis of three-dimensional fracture problems considering thermal effects is presented in this paper. The EDI is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains, temperatures, and derivatives of displacements and temperatures at internal points can be evaluated using the appropriate boundary integral equations. Special emphasis is put on the selection of the auxiliary function that represents the virtual crack advance in the domain integral. This is found to be a key feature to obtain reliable results at the intersection of the crack front with free surfaces. Several examples are analyzed to demonstrate the efficiency and accuracy of the implementation.

scholarly journals Development of boundary-element time-step scheme in solving 3D poroelastodynamics problems

2018 ◽  
Vol 183 ◽  
pp. 01042 ◽  
Author(s):  
Igor Vorobtsov ◽  
Aleksandr Belov ◽  
Andrey Petrov
Keyword(s):  
Boundary Element ◽  
Solid Phase ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Discrete Analogue ◽  
Boundary Integral ◽  
Step Method ◽  
Time Step ◽  
Element Scheme ◽  
Runge Kutta

The development of time-step boundary-element scheme for the three dimensional boundaryvalue problems of poroelastodynamics is presented. The poroelastic continuum is described using Biot’s mathematical model. Poroelastic material is assumed to consist of a solid phase constituting an elastic formdefining skeleton and carrying most of the loading, and two fluid phases filling the pores. Dynamic equations of the poroelastic medium are written for unknown functions of displacement of the elastic skeleton and pore pressures of the filling materials. Green’s matrices and, based on it, boundary integral equations are written in Laplace domain. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. Boundary element scheme is based on time-step method of numerical inversion of Laplace transform. A modification of the time-step scheme on the nodes of Runge-Kutta methods is considered. The Runge-Kutta scheme is exemplified with 2-and 3-stage Radau schemes. The results of comparing the two schemes in analyzing a numerical example are presented.

Sign in / Sign up

Export Citation Format

Share Document