Contact Problems In Elastostatics Research Articles

Capturing Animation-Ready Isotropic Materials Using Systematic Poking

Capturing material properties of real-world elastic solids is both challenging and highly relevant to many applications in computer graphics, robotics and related fields. We give a non-intrusive, in-situ and inexpensive approach to measure the nonlinear elastic energy density function of man-made materials and biological tissues. We poke the elastic object with 3d-printed rigid cylinders of known radii, and use a precision force meter to record the contact force as a function of the indentation depth, which we measure using a force meter stand, or a novel unconstrained laser setup. We model the 3D elastic solid using the Finite Element Method (FEM), and elastic energy using a compressible Valanis-Landel material that generalizes Neo-Hookean materials by permitting arbitrary tensile behavior under large deformations. We then use optimization to fit the nonlinear isotropic elastic energy so that the FEM contact forces and indentations match their measured real-world counterparts. Because we use carefully designed cubic splines, our materials are accurate in a large range of stretches and robust to inversions, and are therefore "animation-ready" for computer graphics applications. We demonstrate how to exploit radial symmetry to convert the 3D elastostatic contact problem to the mathematically equivalent 2D problem, which vastly accelerates optimization. We also greatly improve the theory and robustness of stretch-based elastic materials, by giving a simple and elegant formula to compute the tangent stiffness matrix, with rigorous proofs and singularity handling. We also contribute the observation that volume compressibility can be estimated by poking with rigid cylinders of different radii, which avoids optical cameras and greatly simplifies experiments. We validate our method by performing full 3D simulations using the optimized materials and confirming that they match real-world forces, indentations and real deformed 3D shapes. We also validate it using a "Shore 00" durometer, a standard device for measuring material hardness.

Approximating the influence coefficients of non-planar elastic solids for conformal contact analysis

The exact contact theory is an efficient alternative to the more general yet computationally expensive Finite Element Method for the detailed study of elastostatic contact problems. For its application in conformal contact problems, the exact contact theory needs to be fed with influence coefficients (ICs) appropriate for non-planar solids. An analytical approximation of the ICs for non-planar solids was proposed in a previous work, avoiding the involved process generally necessary to obtain ICs accurately. This work presents further developments of this approximation, further comparison with numerically obtained ICs, and evaluates the errors incurred when using approximated ICs in conformal contact.

On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics

This paper addresses an efficient and robust automatic adaptive local multilevel mesh refinement strategy for unilateral frictional contact problems in elastostatics. The proposed strategy couples the Local Defect Correction multigrid method LDC (Hackbusch, 1984) with the ZZ (Zienkiewicz and Zhu, 1987) a posteriori error estimator. An extension of LDC method to frictional contact problems is introduced. An interesting feature of this extended LDC algorithm is that it still only lies on interpolations of displacement fields. Neither forces conservation nor exchange of contact status is required between the refinement levels. The ZZ a posteriori error estimator is exploited to automatically build the sub-grids of the LDC method. A criterion linked to the relative error in stress is used. The efficiency of the proposed strategy is analyzed on examples derived from nuclear engineering. Practical numerical choices are proposed and justified. The refinement process automatically varies and stops with respect to a given tolerance. Post-treatments show that the sub-grids focus around the contact areas and that the converged LDC solution always respects the prescribed tolerance.

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.

A geophysical application of an elastostatic contact problem

This paper develops an analytical solution to the problem of the interaction between an elastic sealing geologic layer that is embedded within elastically dissimilar geologic media of semi-infinite extent. The axisymmetric mechanics of the elastic sealing layer during the application of pressurization of circular regions located in both halfspace regions is evaluated using an interaction analysis based on the classical theory of elasticity. In particular, the flexural behavior of the elastic sealing layer is modeled as an elastic plate that satisfies the Germain–Poisson–Kirchhoff thin plate theory. Integral expressions are developed for the deflections and stresses in the sealing layer, which are essential for establishing its sealing integrity. Specific numerical results are presented for the case where a single halfspace region is subjected to pressurization over a circular area of finite thickness.

Qualitative analysis of 3D elastostatic contact problems with orthotropic Coulomb friction and solution-dependent coefficients of friction

This paper analyzes a discrete form of 3D contact problems with local orthotropic Coulomb friction and coefficients of friction which may depend on the solution itself. The analysis is based on the fixed-point reformulation of the original problem. Conditions guaranteeing the existence and uniqueness of discrete solutions are established. Finally, numerical results of a model example are presented.

A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics

In this work we consider a stabilized Lagrange (or Kuhn–Tucker) multiplier method in order to approximate the unilateral contact model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed in the convergence analysis. We propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions. Several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method.

Variational formulation and Nonsmooth Optimization Algorithms in Elastostatic Contact Problems for Cracked Body

Variational formulation and Nonsmooth Optimization Algorithms in Elastostatic Contact Problems for Cracked Body

A Stabilized Lagrange Multiplier Method for the Finite Element Approximation of Frictional Contact Problems in Elastostatics

In this work we consider a stabilized Lagrange multiplier method in order to approxi- mate the Coulomb frictional contact model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed. We study the existence and the uniqueness of solution of the discrete problem.

Independent Meshing of Contact Surfaces in 3D Boundary Element Method Contact Analysis

This paper deals with the development of two new boundary element algorithms for solving 3D, frictional, and linear elastostatic contact problems. The main contribution of this research is that solving 3D boundary element models with nonconforming discretizations becomes possible for the first time by using the proposed algorithms. The new algorithms provide the contact constraint equations that will be added to the underdetermined linear system of equations. These algorithms are implemented in a new 3D boundary element code using C++ and verified using several numerical examples. For the models studied, the results using the new boundary element algorithms match well with the finite element results and clearly demonstrate the feasibility of the new boundary element approach for 3D contact analysis.

Uzawa and Newton algorithms to solve frictional contact problems within the bi‐potential framework

AbstractThis paper is concerned with the numerical modeling of three‐dimensional unilateral contact problems in elastostatics with Coulomb friction laws. We propose a Newton‐like algorithm to solve the local contact non‐linear equations within the bi‐potential framework. The piecewise continuous contact tangent matrices are explicitly derived. A comparative study is made between the Newton algorithm and the previously developed Uzawa algorithm. A test example is included to demonstrate the developed algorithms and to highlight their performance. Copyright © 2007 John Wiley & Sons, Ltd.

Convergence of cubic-spline approach to the solution of a system of boundary-value problems

We use cubic spline to derive some consistency relations which are then used to develop a numerical method for the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Boundary formula of order O ( h 8 ) are formulated. The most common approach for convergence analysis are using monotonicity of the coefficient matrix. But here we study a new approach and give the convergence of prescribed method, so that the matrix associated with the system of linear equations that arises, is not required to be monotone. Numerical examples are given to show the applicability and efficiency of our method.

Numerical method for a system of fourth-order boundary value problems

In this paper we use sextic polynomial splines to derive some consistency relations which are then used to develop a numerical method for computing the solution for a system of fourth order boundary-value problems associated with obstacle, unilateral and contact problems. We show that a class of variational inequalities related to contact problems in elasto-statics can be characterized by a sequence of variational equations, which are solved using numerical method. For the purpose of numerical illustrations, we consider example of an elastic beam lying over an elastic obstacle. We have developed the boundary formulas of order four to restrict the band width of the coefficient matrix of the linear system to five. The numerical illustrations and comparison with other methods are given which shows the advantage of the present scheme.

Analysis of 3-D Frictional Contact Mechanics Problems by a Boundary Element Method

The development of two boundary element algorithms for solving 3-D, frictional, and linear elastostatic contact problems is reported in this paper. The algorithms employ nonconforming discretizations for solving 3-D boundary element models, which provide much needed flexibility in the boundary element modeling for 3-D contact problems. These algorithms are implemented in a new 3-D boundary element code and verified using several examples. For the numerical examples studied, the results using the new boundary element algorithms match very well with the results using a commercial finite element code, and clearly demonstrate the feasibility of the new boundary element approach for 3-D contact analysis.

A generalized adaptive incremental approach for solving inequality problems of convex nature

A proposed incremental model for the solution of a general class of convex programming problems is introduced. The model is an extension of that developed by Mahmoud et al. (1993) which is limited to linear constraints having nonzero free coefficients. In the present model, this limitation is relaxed, and allowed to be zero. The model is extended to accommodate those constraints of zero free coefficients. The proposed model is applied to solve the elasto-static contact problems as a class of variation inequality problems of convex nature. A set of different physical nature verification examples is solved and discussed in this paper.

Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary-Value Problems

In this paper, we use parametric quintic splines to derive some consistency relations which are then used to develop a numerical method for computing the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Numerical evidence is presented to show the applicability and superiority of the new method over other collocation, finite difference, and spline methods.

BEM and FEM analysis of Signorini contact problems with friction

In this paper, we present a comparative study of the boundary element method (BEM) and the finite element method (FEM) for analysis of Signorini contact problems in elastostatics with Coulomb's friction law. Particularities of each method and comparison with the penalty method are discussed. Numerical examples are included to demonstrate the present formulations and to highlight its performance.

Stationary Damage Concept For PoroelasticContact Mechanics

This paper proposes the concept of stationary damage of the porous fabric of a poroelastic material as a means of examining the transient processes that occur in contact problems associated with poroelastic media. This approach combines the advantages of classical elastostatic contact problems in identifying the extent of initial damage during indentation of a fluid saturated porous medium. The stationary damage concept transforms the initially homogeneous medium to one that is both elastically and hydraulically inhomogeneous. Standard computational procedures can then be applied to solve the resulting poroelastic contact problem.

Study of variational inequality and equality formulations for elastostatic frictional contact problems

An overview ofvariational inequality andvariational equality formulations for frictionless contact and frictional contact problems is provided. The aim is to discuss the state-of-the-art in these two formulations and clearly point out their advantages and disadvantages in terms of mathematical completeness and practicality. Various terms required to describe the contact configuration are defined.Unilateral contact law and classical Coulomb’s friction law are given.Elastostatic frictional contact boundary value problem is defined. General two-dimensional frictionless and frictional contact formulations for elastostatic problems are investigated. An example problem of a two bar truss-rigid wall frictionless contact system is formulated as an optimization problem based on the variational inequality approach. The problem is solved in a closed form using the Karush-Kuhn-Tucker (KKT) optimality conditions. The example problem is also formulated as a frictional contact system. It is solved in the closed form using a new two-phase analytical procedure. The procedure avoids use of the incremental/iterative techniques and user defined parameters required in a typical implementation based on the variational equality formulation. Numerical solutions for the frictionless and frictional contact problems are compared with the results obtained by using a general-purpose finite element program ANSYS (that uses variational equality formulation). ANSYS results match reasonably well with the solutions of KKT optimality conditions for the frictionless contact problem and the two-phase procedure for the frictional contact problem. The validity of the analytical formulation for frictional contact problems (with one contacting node) is verified. Thevariational equality formulation for frictionless and frictional, contact problems is also studied in detail. The incremental/iterative Newton-Raphson scheme incorporating the penalty approach is utilized. Studies are conducted to provide insights for the numerical solution techniques. Based on the present study it is concluded that alternate formulations and computational procedures need to be developed for analysis of frictional contact problems.

Transient elastodynamics around cracks including contact and friction

In this paper, some numerical investigations of fast transient elastodynamic problems are presented, which consider especially the interaction of elastic waves with cracks, including contact and friction. For this purpose, a boundary element formulation in the time domain is coupled with contact mechanics models previously proposed and tested by the authors in the framework of elastostatic contact problems. In the numerical examples, qualitative and quantitative effects of the interaction between elastic waves and non-classical, unilateral cracks are briefly discussed.