Elastic Contact Problems Research Articles

On R-linear convergence of semi-monotonic inexact augmented Lagrangians for bound and equality constrained quadratic programming problems with application

New convergence results for a variant of the inexact augmented Lagrangian algorithm SMALBE [Z. Dostál, An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum, Computing 78 (2006) 311–328] for the solution of strictly convex bound and equality constrained quadratic programming problems are presented. The algorithm SMALBE-M presented here uses a fixed regularization parameter and controls the precision of the solution of auxiliary bound constrained problems by a multiple of the norm of violation of the equality constraints and a constant which is updated in order to enforce the increase of Lagrangian function. A nice feature of SMALBE-M is its capability to find an approximate solution of important classes of problems in a number of iterations that is independent of the conditioning of the equality constraints. Here we prove the R-linear rate of convergence of the outer loop of SMALBE-M for any positive regularization parameter after the strong active constraints of the solution are identified. The theoretical results are illustrated by solving two benchmarks, including the contact problem of elasticity discretized by two million of nodal variables. The numerical experiments indicate that the inexact solution of auxiliary problems in the inner loop results in a very small increase of the number of outer iterations as compared with the exact algorithm. The results do not assume independent equality constraints and remain valid when the solution is dual degenerate.

Contact with a Corner for Nonlinearly Elastic Rods

In elastic contact problems it is usually required that the contact force has to be directed normally to the contact surface in the absence of friction. For an obstacle with nonsmooth surface this gives infinitely many normal directions at an edge or at a corner. For the case where a nonlinearly elastic rod under terminal loads is hanging over a needle, it is shown that the balance equations supplemented with such a normality condition have a continuum of solutions. Moreover, an additional contact condition is derived from a corresponding variational problem by means of special inner variations that preserve the shape of the rod. This way one is finally lead to a unique solution at least locally.

The use of static reduction in the finite element solution of two-dimensional frictional contact problems

In this paper, detailed instructions are given for performing static reduction on a finite element description of an elastic contact problem, thus reducing the dimensionality of the problem to the set of contact nodes alone. This significantly reduces the computational time for the solution to evolutionary contact problems and also gives the user greater control over the detailed implementation of the contact and friction laws. The reduced stiffness matrix is also an essential ingredient in the determination of the critical coefficient of friction for the problem to be well posed, and it facilitates the determination of the conditions under which a frictional system may shake down under periodic loading.

A new solver for the elastic normal contact problem using conjugate gradients, deflation, and an FFT-based preconditioner

This paper presents our new solver BCCG+FAI for solving elastic normal contact problems. This is a comprehensible approach that is based on the Conjugate Gradients (CG) algorithm and that uses FFTs. A first novel aspect is the definition of the “FFT-based Approximate Inverse” preconditioner. The underlying idea is that the inverse matrix can be approximated well using a Toeplitz or block-Toeplitz form, which can be computed using the FFT of the original matrix elements. This preconditioner makes the total number of CG iterations effectively constant in 2D and very slowly increasing in 3D problems. A second novelty is how we deal with a prescribed total force. This uses a deflation technique in such a way that CGs convergence and finite termination properties are maintained. Numerical results show that this solver is more effective than existing CG-based strategies, such that it can compete with Multi-Grid strategies over a much larger problem range. In our opinion it could be the new method of choice because of its simple structure and elegant theory, and because robust performance is achieved independently of any problem specific parameters.

Total FETI domain decomposition method and its massively parallel implementation

We describe an efficient massively parallel implementation of our variant of the FETI type domain decomposition method called Total FETI with a lumped preconditioner. A special attention is paid to the discussion of several variants of parallelization of the action of the projections to the natural coarse grid and to the effective regularization of the stiffness matrices of the subdomains. Both numerical and parallel scalability of the proposed TFETI method are demonstrated on a 2D elastostatic benchmark up to 314,505,600 unknowns and 4800cores. The results are also important for implementation of scalable algorithms for the solution of nonlinear contact problems of elasticity by TFETI based domain decomposition method.

Approximation results and subspace correction algorithms for implicit variational inequalities

This paper deals with the mathematical analysis and the subspace approximation of a system of variational inequalities representing a unified approach to several quasistatic contact problems in elasticity. Using an implicit time discretization scheme and some estimates, convergence properties of the incremental solutions and existence results are presented for a class of abstract implicit evolution variational inequalities involving a nonlinear operator. To solve the corresponding semi-discrete and the fully discrete problems, some general subspace correction algorithms are proposed, for which global convergence is analyzed and error estimates are established.

Design and Application of Hydraulic-Walking Incremental Launching Equipment

This work designs a novel hydraulic-walking incremental launching equipment. The working principle of the equipment is proposed. Considering that the contact in incremental launching construction is a typical elastic contact problem, and changes in contact state will directly affect the stability of the incremental launching, this work proposes the equilibrium equations of computational contact mechanics and numerical analysis method for the incremental launching equipment through analyzing the working process of the equipments. The proposed model and method are applied in the analysis of typical working conditions of the incremental launching of Donghua Bridge. The structural characteristics in the construction process are analyzed and discussed. The analysis results show that the strength and stability of the steel box girder can meet the construction requirements. The incremental launching of Donghua Bridge was completed successfully. The proposed model and method can be used to predict and analyze the structural performance in incremental launching construction as well as provide theoretical reference and analysis method for similar constructions.

3D reconstruction of TMJ after resection of the cyst and the stress–strain analyses

The aim of this article is 3D analysis of the temporomandibular joint (TMJ) patient, who underwent surgery, during which the right TMJ was resected along with the ramus of mandible and consequently the joint was reconstructed with subtotal replacement. The main goal is to give a suitable formulation of mathematical model, which describes the changes of stresses in TMJ incurred after the surgery. The TMJ is a complex, sensitive and highly mobile joint which works bilaterally so each side influences the contralateral joint and because of this the distribution of the stresses is changed in the healthy joint as well. Detailed knowledge about function these are necessary for clinical application of temporomandibular joint prosthesis and also help us estimate the lifetime of the prosthesis a possibilities of alteration in the contra lateral joint components. The geometry for the 3D models is taken from the CT scan date and its numerical solution is based on the theory of semi-coercive unilateral contact problems in linear elasticity. This article provides medical part with case report, discretion of treatment, than the methods of mathematical modeling and his possibilities are described and finally results are reported.

Study on teeth profile modification of cycloid reducer based on non-Hertz elastic contact analysis

To solve the problem of contact failure for heavy-duty cycloid reducers, the teeth mesh was originally studied as a Non-Hertz elastic contact problem. An analysis method called non-Hertz flexibility matrix method (NHFMM) for the teeth contact was developed for the teeth profile modification of pin gear. The NHFMM analysis shows that the edge concentrated pressure calculated by non-Hertz method is 2.11 times of that by Hertz method on the pin teeth without profile modification. In the end, the analysis results of NHFMM were verified by a specially designed photo-elastic experiment.

Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity

In this paper, some new generalized Newton’s methods for the resolution of elastostatic frictional contact problem approximated by finite elements are presented and compared to existing ones. A numerical experimentation is performed to compare the different methods, especially with respect to the sensitivity to the method parameter. Two different strategies to approximate the contact and friction condition are considered: a nodal and an integral one. Existence and uniqueness results of the solution to the discretized problem are also discussed.

Finite-Element-Based Multiple Normal Loading-Unloading of an Elastic-Plastic Spherical Stick Contact

The repeated normal elastic plastic contact problem of a deformable sphere against a rigid flat under full stick contact condition is investigated with a commercial finite element software ANSYS. Emphasis is placed on the effect of strain hardening and hardening model with the maximum interference of load ranging from elastic to fully plastic, which has not yet been reported. Different values of tangent modulus coupled with isotropic and kinematic hardening models are considered to study their influence on contact parameters. Up to ten normal loading-unloading cycles are applied with a maximum interference of 200 times the interference required to initiate yielding. Results for the variation of mean contact pressure, contact load, residual interference, and contact area with the increasing number of loading unloading cycles at high hardening parameter as well as for low tangent modulus with two different hardening models are presented. Results are compared with available finite element simulations and in situ results reported in the literature. It is found that small variation of tangent modulus results in same shakedown behavior and similar interfacial parameters in repeated loading-unloading with both the hardening rules. However at high tangent modulus, the strain hardening and hardening rules have strong influence on contact parameters.

An explicit solution for the elastic quarter-space problem in matrix formulation

This paper presents a fast and convenient algorithm for the solution of the elastic quarter-space contact problem, which uses discretization to form matrices to realize the overlapping solution process for the elastic quarter-space as developed by Hetenyi [Hetenyi, M., 1970. A general solution for the elastic quarter space, Trans. ASME Journal of Applied Mechanics 37 E(1), 70–76]. This proposed method provides an explicit solution which is as yet absent in existing literatures. The generated matrices are only related to the mesh structure and Poisson’s ratio while unrelated to the loading, such that they can be applied to different loading cases. Hence, the present method offers a possibility for substantially improving the efficiency of those numerical iterative analyses, such as the elastohydrodynamic lubrication of the contact in an elastic quarter-space. Verification of the present method was accomplished by comparison with the existing quarter-space results.

Normal contact between a rigid surface and a viscous body: Verification of the method of reduction of dimensionality for viscous media

Analogies between elastic and viscous contact problems are used to apply boundary element algorithms and the method of reduction of dimensionality respectively to the problem of normal contact between a rigid body and a viscous half space. Some basic examples are examined in order to compare both methods and their results to each other.

Investigation of the dry normal contact between fractal rough surfaces using the reduction method, comparison to 3D simulations

The reduction method was applied to solving the elastic contact problem of fractal rough surfaces having different fractal dimensions. These surfaces are of particular interest, as contact spots influence each other on large and small scales, depending on the power spectrum. Results were compared to previous boundary-element method simulations that had been conducted with 3D surfaces, taking into account all interdependencies between contact spots. Excellent agreement in the contact stiffness was found. Even though no interactions are allowed in the reduced model, the same power-law behavior can be observed as in the 3D case, due to the transformation of the power spectrum. Theoretical considerations about the power-law for different fractal dimensions are given.

Scalable TFETI with optional preconditioning by conjugate projector for transient frictionless contact problems of elasticity

The FETI based domain decomposition method is adapted to implement the time step of the Newmark scheme for the solution of dynamic contact problems without friction. If the ratio of the decomposition and discretization parameters is kept uniformly bounded, then the cost of the time step is proved to be proportional to the number of nodal variables. The algorithm uses our in a sense optimal MPRGP algorithm for the solution of strictly convex bound constrained quadratic programming problems with optional preconditioning by the conjugate projector to the subspace defined by the trace of the rigid body motions on the artificial subdomain interfaces. The proof of optimality combines the convergence theory of our MPRGP algorithm, the classical bounds on the spectrum of the mass and stiffness matrices, and our theory of the preconditioning by a conjugate projector for nonlinear problems. The results are confirmed by numerical solution of 2D and 3D dynamic contact problems.

A Nitsche-Based Method for Unilateral Contact Problems: Numerical Analysis

We introduce a Nitsche-based finite element discretization of the unilateral contact problem in linear elasticity. It features a weak treatment of the nonlinear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the $H^1(\Omega)$-norm for linear finite elements in two dimensions, which is $O(h^{\frac{1}{2}+\nu})$ when the solution lies in $H^{\frac{3}{2}+\nu}(\Omega)$, $0<\nu\leq 1/2$. An interest of the formulation is that, as opposed to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf-sup condition needs to be satisfied.

An interior-point algorithm for the minimization arising from 3D contact problems with friction

The paper deals with a variant of the interior-point method for the minimization of strictly quadratic objective function subject to simple bounds and separable quadratic inequality constraints. Such minimizations arise from the finite element approximation of contact problems of linear elasticity with friction in three space dimensions. The main goal of the paper is the convergence analysis of the algorithm and its implementation. The optimal preconditioners for solving ill-conditioned inner linear systems are proposed. Numerical experiments illustrate the computational efficiency for large-scale problems.

A Gradient-Less Shape Optimization for Elastic Contact Problems

A shape optimization method for solving elastic contact problems is proposed based on gradient-less idea(a zero order optimization method). It is found that the contact stress on the elastic contact surface can be smoothed and reduced effectively by both changing the contact surface shape and changing the shape of non-contact surface. To avoid mesh distortion that results in undesirable shape, a move stratrgy for nodal movement has been used in the finite element model. Two examples are presented to show the effectiveness of the proposed method in dealing with elastic contact problems.

Topology optimization of quasistatic contact problems

Topology optimization of quasistatic contact problemsThis paper deals with the formulation of a necessary optimality condition for a topology optimization problem for an elastic contact problem with Tresca friction. In the paper a quasistatic contact model is considered, rather than a stationary one used in the literature. The functional approximating the normal contact stress is chosen as the shape functional. The aim of the topology optimization problem considered is to find the optimal material distribution inside a design domain occupied by the body in unilateral contact with the rigid foundation to obtain the optimally shaped domain for which the normal contact stress along the contact boundary is minimized. The volume of the body is assumed to be bounded. Using the material derivative and asymptotic expansion methods as well as the results concerning the differentiability of solutions to quasistatic variational inequalities, the topological derivative of the shape functional is calculated and a necessary optimality condition is formulated.

Study on teeth profile modification of cycloid reducer based on non-Hertz elastic contact analysis

To solve the problem of contact failure for heavy-duty cycloid reducers, the teeth mesh was originally studied as a non-Hertz elastic contact problem. An analysis method called non-Hertz flexibility matrix method (NHFMM) for the teeth contact was developed for the teeth profile modification of pin gear. The NHFMM analysis shows that the edge concentrated pressure calculated by non-Hertz method is 2.11 times of that by Hertz method on the pin teeth without profile modification. In the end, the analysis results of NHFMM were verified by a specially designed photo-elastic experiment.