General Interface Integral Equations in Elasticity of Random Structure Composites

One considers a linear elastic composite material (CM, [1]), which consists of a homogeneous matrix containing the random set of heterogeneities. An operator form of the general integral equation (GIE, [2–6]) connecting the stress and strain fields in the point being considered and the surrounding points are obtained for the random fields of inclusions in the infinite media. The new GIE is presented in a general form of perturbations introduced by the heterogeneities and defined at the inclusion interface by the unknown fields of both the displacement and traction. The method of obtaining of the GIE is based on a centering procedure of subtraction from both sides of a new initial integral equation their statistical averages obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known centering methods. One proves the absolute convergence of the proposed GIEs, and some particular cases, asymptotic representations, and simplifications of proposed GIEs are presented for the particular constitutive equations of linear thermoelasticity. In particular, we use a meshfree method [7] based on fundamental solutions basis functions for a transmission problem in linear elasticity. Numerical results were obtained for 2D CMs reinforced by noncanonical inclusions.

A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of inclusions, when the concentration of the inclusions is a function of the coordinates (so-called functionally graded materials). The composite medium is subjected to essentially inhomogeneous loading by the fields of the stresses, temperature, and body forces (e.g. for a centrifugal load). The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained for the random fields of inclusions. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions, such as effective field hypothesis implicitly exploited in the known centering methods. In so doing the size of a region including the inclusions acting on a separate one is finite, i.e. the locality principle takes place.

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing either a statistically inhomogeneous random or deterministic set of inclusions, when the concentration of the inclusions is a function of the coordinates (so-called functionally graded materials). The composite medium is subjected to essentially inhomogeneous loading by the fields of the stresses, temperature, and body forces (e.g., for a centrifugal load). The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained for the random and deterministic fields of inclusions. In so doing, the size of a region including the inclusions acting on a separate one is finite, that is, the locality principle takes place.

In this paper, mathematical properties of variational solutions and boundary integral equation solutions of linear elasticity problems are studied. It is first noted that a variational solution of the three-dimensional linear elasticity problem exists in the Sobolev space [H1(Ω)]3. It is then shown that a unique solution of the boundary integral equation is identical to the variational solution in [H1(Ω)]3. To represent the boundary integral equation, Green's formula in the Sobolev space is utilized on the solution domain, excluding a ball, with small radius ρ that is centered at the point where the point load is applied. By letting ρ tend to zero, it is shown that, for the linear elasticity problem, the boundary integral expression is valid for the variational solution. From this fact, a numerical approximation of the variational solution can be obtained by the boundary element method, even when a classical solution does not exist.

The article presents a novel model order reduction method for mechanical problems in linear elasticity with nonlinear contact conditions. Recently, we have proposed an efficient reduction scheme for the node‐to‐node formulation [Manvelyan et al., Comput Mech 68, 1283–1295 (2021)] that leads to linear complementarity problems (LCPs). Here, we enhance the underlying contact problem to a node‐to‐segment formulation, which leads to quadratic inequalities as constraints. The adjoint system corresponds to a nonlinear complementarity problem that describes the Lagrange multiplier. The latter is solved by a Newton‐type iteration based on a LCP solver in each time step. Since the maximal set of potential contact nodes is predefined, an additional substructuring by Craig–Bampton can be performed. This contact treatment turns out to be necessary and allows exclusion of the Lagrange multipliers and the nodal displacements at contact from reduction. The numerical solutions of the reduced contact problem achieve high accuracy and the dynamic contact carries over the behavior of the full order model. Moreover, if the contact area is small compared to the overall structure, the reduction scheme performs very efficiently. The performance of the resulting reduction method is assessed on two 2D computational examples from linear elasticity.

Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity

In this paper, we are interested in solving a Cauchy inverse problem in linear elasticity. For this, we propose a new method based on Robin conditions on the inaccessible boundary, then we study the convergence and regularizing property of the proposed algorithm. We use the finite element method for the discretization of our problem. Further, we treat the spectrum analysis of the discrete problem in order to study the convergence behavior of the proposed method in the discrete case. Finally, we present numerical results which show the efficiency and stability of the proposed approach in the presence of perturbed data. The robustness of the proposed algorithm is also performed for the anisotropic and heterogeneous cases.

We derive globally convergent multigrid methods for the discretized Signorini problem in linear elasticity. Special care has to be taken in the case of spatially varying normal directions. In numerical experiments for 2 and 3 space dimensions we observed similar convergence rates as for corresponding linear problems.

For a dual-dual formulation of a frictional contact problem in linear elasticity we present a mixed finite element method based on PEERS elements and we analyze the performance of a nested Uzawa algorithm.

Boundary augmented Lagrangian method for contact problems in linear elasticity

In this paper a mixed least squares finite element method for solving problems in linear elasticity is proposed. The developed numerical technique allows the use of separate unknowns for displacements and stresses, discontinuous interpolation functions for displacements, and the resulting linear system has a symmetric and positive definite coefficient matrix. The approximate solution of the linear elasticity problem is obtained by minimization of a least squares functional based on the constitutive equations and equations of equilibrium. The proposed method is implemented in an original computer code written in C programming language. Its performance is tested on classical examples from theory of elasticity with well-known exact analytical solutions. Results from the implementation of a constant displacement-bilinear stress element and bilinear displacement-bilinear stress element are discussed.

Painlevé analysis of nonlinear evolution equations—an algorithmic method

In a recent paper by Fayolle, Mitrani, and Iasnogorodski [2], some general multidimensional integral equations were derived in order to solve for the mean response time of each of several classes in a queue whose service discipline was weighted processor sharing. The arrival processes were Poisson. The weighting means that each job within a class k is given an amount of processing proportional to the priority weight gk associated with that class. For exponential service times, the general equations were solved. In this note, a simple observation allows use of the exponential solution directly for the case of hyperexponential servers. As a result, it is possible to state the following. •Characterization of a server in terms of its mean and coefficient of variation is not sufficient to predict even the mean response time for a class using weighted processor sharing. In unweighted or egalitarian processor sharing, only the mean is sufficient. •The Kleinrock conservation law [4] does not hold for nonexponential servers. Fayolie et al. [2] had showed that it did hold for exponential servers.

On bilinear representations and infinite conservation laws of a nonlinear variable-coefficient equation