Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems

The traditional method of fundamental solutions (MFS) based on the “global” boundary discretization leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. In this study, a localized version of the MFS (LMFS) is proposed for the large-scale modeling of two-dimensional (2D) elasticity problems. In the LMFS, the whole analyzed domain can be divided into small subdomains with a simple geometry. To each of the subdomain, the traditional MFS formulation is applied and the unknown coefficients on the local geometric boundary can be calculated by the moving least square method. The new method yields a sparse and banded matrix system which makes the method very attractive for large-scale simulations. Numerical examples with up to 200,000 unknowns are solved successfully using the developed LMFS code.     

Solution of the fundamental problems of elasticity theory for a multiconnected anisotropic half-plane

By analytic continuation of generalized complex potentials to upper half-planes we reduce the boundary conditions on a rectilinear boundary to problems of linear coupling for cuts of a multiconnected extended plane. By solving the latter problems we obtain general representations of the complex potentials in the case of a multiconnected anisotropic half-plane for different types of boundary conditions on intervals of the rectilinear boundary. As particular cases we give expressions for the complex potentials in the cases of action of external forces on the rectilinear boundary and dies both with and without friction. Two figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 44–63.     

On some representations of solutions of dynamical problems in elasticity theory

The divergence and rotation of a translation vector field, which are solutions of wave equations, are used as auxiliary potentials for numerical solution of dynamical problems in elasticity theory. In a particular case, the problems of longitudinal translation are tested in two variants of numerical algorithms with their programming realization, some suggestions are given regarding the choice of the optimal algorithm for achieving highly precise calculations with a small number of discretization points.Translated from Dinamicheskie Sistemy, No. 9, pp. 27–33, 1990.     

Finite element construction on the basis of analytic solutions of elasticity theory problems

A finite element in the form of a part of a circular ring loaded by a uniformly distributed radial load is constructed on the basis of the exact solution of an elasticity theory problem. This permits a system of resolving equations to be obtained by the finite element method with a three-diagonal matrix. An example is presented that illustrates the accuracy and efficiency of utilization of the elements mentioned.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 85–90, 1988.     

Solution of the fundamental problems of the theory of elasticity for a half-plane with holes and cracks

We obtain the general solution of the fundamental problems of the theory of elasticity for an isotropic half-plane with a finite number of arbitrarily situated elliptic holes whose boundaries may intersect or form rectilinear cuts or boundaries of curvilinear holes. On the rectilinear boundary the first problem and the second or mixed problem of the theory of elasticity are defined. We use general expressions obtained previously by the author for the complex potentials generated by solving the problem of linear coupling for cuts in a multiconnected region, conformal mappings, and the method of least squares. The problem is reduced to solving a system of linear algebraic equations. The results of numerical experiments are given for a half-plane with a crack in the case of the first fundamental problem and the action of various loads. Two figures, two tables. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 157–171.     

Solution of some two-dimensional problems of elasticity

Using the method of boundary elements, we obtain numerical solutions of two-dimensional (plane deformation) boundary-value problems on the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

Error estimates for certain approximate solutions of problems in the linear theory of elasticity

Zusammenfassung Eine Näherungslösung eines linearen Elastizitätsproblems kann manchmal erreicht werden mittels einer (kleinen) Änderung des Ausdrucks für die elastische Energie. Obere Schranken für die Fehler einer solchen Näherungslösung werden abgeleitet. Das Ergebnis hat auch Bedeutung vom grundsätzlichen Gesichtspunkt, weil die «exakte» Form des Energieausdrucks für wirkliche elastische Systeme niemals genau bekannt ist.

---

---

Dedicated to Professor Dr. Hans Ziegler on the occasion of his 60th birthday     

Biorthogonal expansions in the first fundamental problem of elasticity theory

The first fundamental boundary-value problem of elasticity theory is considered for a rectangular semi-infinite strip whose long sides are free of stress. Separation of variables is used to reduce the solution to a series expansion of two functions defined in a closed interval (the “end” of the half-strip), in terms of homogeneous solutions. The system of homogeneous solutions over an interval of the real axis is proved to be complete in L₂. Systems of functions biorthogonal to the systems of homogeneous solutions are constructed on a certain contour on the Riemann surface of the logarithm. This biorthogonality concept is a natural generalization of biorthogonality over a closed interval. The biorthogonal systems constructed are used to find explicit expressions for the expansion coefficients.     

A comparison of the solutions of the problems of the theory of shells and the three-dimensional theory of elasticity

By comparing with the results obtained by numerical solution of a three-dimensional problem of the theory of elasticity we evaluate the different versions of the theory of elastic orthotropic cylindrical shells.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 96–101.     

Relaxation of variational problems in two-dimensional nonlinear elasticity

Consider a plate occupying in a reference configuration a bounded open set Ω ⊂ ℝ ² , and let be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:

Estimates for deviations from exact solutions to plane problems in the cosserat theory of elasticity

A functional a posteriori estimate is obtained for control of the accuracy of approximate solutions to plane problems arising in the Cosserat theory of elasticity. We deal with the case where displacements and independent rotation are given on the boundary of the domain, a continuous medium is isotropic, and there is a linear dependence between stresses and strains. The proposed method is based on the duality theory of Calculus of Variations and can be also applied to the case of anisotropic media.     

Some fundamental problems of the theory of thermoviscoelasticity

Methods of solving quasistatic problems of the linear theory of thermoviscoelasticity are discussed. Special attention is given to the method of approximations. Modern methods of solving contact problems with a variable boundary and problems with a time-dependent boundary are briefly reviewed. Methods of solving problems with a nonuniform temperature field are outlined. The basic relations of the nonlinear theory of thermoviscoelasticity are given. Its various modifications and simplifications associated with the introduction of fairly general assumptions are examined. Methods of solving problems in certain nonlinear theories are noted. Nonisothermal (coupled) problems of thermoviscoelasticity and questions relating to the general theory with physical and geometric nonlinearity are discussed.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 41–58, January–February, 1971.     

The first and second fundamental boundary-value problems of elasticity theory for a transversally isotropic paraboloid of revolution

We give an exact solution of the interior and exterior problems of elasticity theory for a transversally isotropic paraboloid of revolution in the case when the stresses prescribed on its surface or the displacements along one variable can be represented by a Hankel integral and those along the other variable can be expanded in a trigonometric series. It is assumed that the roots of the characteristic equation are of multiplicity greater than one.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 60–70.     

Homogeneous solutions in three-dimensional problems of the theory of elasticity for transtropic plates with restrained ends

Homogeneous solutions of equations of the theory of elasticity for transtropic plates, on the flat faces of which the components of the displacement vector are zero, are obtained. The solution consists of the turbulent and potential states of the plate.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 10–13, 1988.     

Spatially periodic fundamental solutions of the theory of oscillations

Spatially periodic fundamental solutions of the theory of oscillations are constructed, applicable to anisotropic elastic media with a general form of anisotropy. The results are compared with the isotropic case.     

Functional a posteriori estimates of the error in the solutions of plane problems in Cosserat elasticity theory

A posteriori estimates are obtained that enable one to control the accuracy of conformal approximations of the solution of plane problems arising in the theory of Cosserat elasticity, irrespective of the method of constructing these approximations. Problems are considered in which the boundary conditions can include both specified displacements as well as surface forces and torques.     

Homogenization of two-dimensional elasticity problems with very stiff coefficients

In this paper we study the asymptotic behaviour of a sequence of two-dimensional linear elasticity problems with equicoercive elasticity tensors. Assuming the sequence of tensors is bounded in L¹, we obtain a compactness result extending to the elasticity the div-curl approach of [M. Briane, J. Casado-Díaz, Two-dimensional div-curl results. Application to the lack of nonlocal effects in homogenization, Comm. Partial Differential Equations 32 (2007) 935-969] for the conduction. In the periodic case this compactness result is refined replacing the L¹-boundedness by a less restrictive condition involving the oscillations period. We also build a sequence of isotropic elasticity problems with L¹-unbounded Lamé's coefficients, which converges to a second gradient limit problem. This loss of compactness shows a gap in the limit behaviour between the very stiff problems of elasticity and those of conduction. Indeed, in the conduction case a compactness result was proved in [M. Briane, J. Casado-Díaz, Asymptotic behaviour of equicoercive diffusion energies in dimension two, Calc. Var. Partial Differential Equations 29 (4) (2007) 455-479] without assuming any bound from above for the conductivities.     

Application of singular integral equations in two-dimensional problems of the theory of elasticity for piecewise-uniform bodies with cracks

Using the method of singular integral equations we solve a two-dimensional problem of the theory of elasticity for an infinite plate containing an elastic inclusion of arbitrary configuration and a system of curvilinear incisions. The numerical solution is found by the method of mechanical quadratures for the case of an elliptic inclusion and a single polygonal crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 93–98.     

Asymptotic behavior of solutions to problems of elasticity theory at infinity in flat parabolic domains

Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem on the unique solvability of the problem in spaces of the type L₂ containing degrees of distance r=|x| as weight multipliers. For the Neumann conditions (stresses are prescribed on the boundary of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the last problem is simplified. The existence theorems and conditions for solvability of the “elastic” Neumann problem are presented. These results are based on the weighted Korn inequality. Bibliography: 29 titles. Translated fromProblemy Matematicheskogo Analiza. No. 15, 1995, pp. 162–200