Contact Mechanics Problems Research Articles

Numerical homogenization of fractal interface problems

We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal geometry. Our main results concern the construction of projection operators with suitable stability and approximation properties. The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD) and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems.

Dynamic frictional contact mechanics between a functionally graded orthotropic medium and a moving flat punch

The present study establishes a general theory for dynamic frictional contact mechanics between an orthotropic graded half-plane and a moving rigid punch of a flat profile. The rigid punch moves over the orthotropic graded half-plane at a constant subsonic speed. Analytical method is developed for the contact mechanics problem based on two-dimensional elastodynamics. Stresses in the orthotropic graded medium are determined considering plane elasticity. Governing partial differential equations are solved analytically by the use of Galilean and Fourier transformations. The unknown functions appearing in the displacement components are determined considering the surface boundary conditions. The mixed boundary value problem is then reduced to a singular integral equation of the second kind which is solved numerically using an appropriate expansion-collocation technique. The contact stress results of the present analytical study are compared to those generated through a finite element analysis regarding the elastostatic case. Moreover, the contact stress results obtained in the elastodynamic case are also verified using some analytical results available in the literature. Successful agreement is attained in all the comparative analyses. Normal contact stress, lateral contact stress and punch stress intensity factors are calculated for various values of punch speed, coefficient of friction, in-homogeneity constant and orthotropic elastic constants. Results of the present study clearly indicates that the utilization of elastodynamic theory is crucial to get more realizable and accurate estimations of contact stresses especially in high speed sliding conditions.

Deep solution operators for variational inequalities via proximal neural networks

Following Bauschke and Combettes (Convex analysis and monotone operator theory in Hilbert spaces, Springer, Cham, 2017), we introduce ProxNet, a collection of deep neural networks with ReLU activation which emulate numerical solution operators of variational inequalities (VIs). We analyze the expression rates of ProxNets in emulating solution operators for variational inequality problems posed on closed, convex cones in real, separable Hilbert spaces, covering the classical contact problems in mechanics, and early exercise problems as arise, e.g., in valuation of American-style contracts in Black–Scholes financial market models. In the finite-dimensional setting, the VIs reduce to matrix VIs in Euclidean space, and ProxNets emulate classical projected matrix iterations, such as projected Jacobi and projected SOR methods.

A survey of numerical methods for hemivariational inequalities with applications to contact mechanics

In this paper we present an abstract nonsmooth optimization problem for which we recall existence and uniqueness results. We show a numerical scheme to approximate its solution. The theory is later applied to a sample static contact problem that describes an elastic body in frictional contact with a foundation. This problem leads to a hemivariational inequality which we solve numerically. Finally, we compare three computational methods of solving contact mechanical problems: the direct optimization method, the augmented Lagrangian method and the primal–dual active set strategy.

A scalable preconditioning framework for stabilized contact mechanics with hydraulically active fractures

A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. We focus on a blended finite element/finite volume method, where the porous medium is discretized by low-order continuous finite elements with nodal unknowns, cell-centered Lagrange multipliers are used to prescribe the contact constraints, and the fluid flow in the fractures is described by a classical two-point flux approximation scheme. This formulation is consistent, but is not uniformly inf-sup bounded and requires a stabilization. For the resulting 3×3 block Jacobian matrix, robust and efficient solution methods are not available, so we aim at designing new scalable preconditioning strategies based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid techniques. The key idea is to restrict the system to a single-physics problem, approximately solve it by an inner algebraic multigrid approach, and finally prolong it back to the fully-coupled problem. Two different techniques are presented, analyzed and compared by changing the ordering of the restrictions. Numerical results illustrate the algorithmic scalability, the impact of the relative number of fracture-based unknowns, and the performance on a benchmark problem. The objective of the analysis is to identify the most promising solution strategy.

Condition Number and Clustering-Based Efficiency Improvement of Reduced-Order Solvers for Contact Problems Using Lagrange Multipliers

This paper focuses on reduced-order modeling for contact mechanics problems treated by Lagrange multipliers. The high nonlinearity of the dual solutions lead to poor classical data compression. A hyper-reduction approach based on a reduced integration domain (RID) is considered. The dual reduced basis is the restriction to the RID of the full-order dual basis, which ensures the hyper-reduced model to respect the non-linearity constraints. However, the verification of the solvability condition, associated with the well-posedness of the solution, may induce an extension of the primal reduced basis without guaranteeing accurate dual forces. We highlight the strong link between the condition number of the projected contact rigidity matrix and the precision of the dual reduced solutions. Two efficient strategies of enrichment of the primal POD reduced basis are then introduced. However, for large parametric variation of the contact zone, the reachable dual precision may remain limited. A clustering strategy on the parametric space is then proposed in order to deal with piece-wise low-rank approximations. On each cluster, a local accurate hyper-reduced model is built thanks to the enrichment strategies. The overall solution is then deeply improved while preserving an interesting compression of both primal and dual bases.

Tykhonov well-posedness of fixed point problems in contact mechanics

We consider a fixed point problem mathcal {S}u=u where {mathcal {S}:C(mathbb{R}_{+};X)to C(mathbb{R}_{+};X)} is an almost history-dependent operator. First, we recall the unique solvability of the problem. Then, we introduce the concept of Tykhonov triple, provide several relevant examples, and prove the corresponding well-posedness results for the considered fixed point problem. This allows us to deduce various consequences which illustrate the stability of the solution with respect to perturbations of the operator mathcal {S}. Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in solid and contact mechanics. To give some examples, we consider two mathematical models which describe the equilibrium of a viscoelastic body in frictionless contact with a foundation. We state the mechanical problems, list the assumptions on the data, and derive their associated fixed point formulation. Then, we illustrate the use of our abstract results in order to deduce the continuous dependence of the solution with respect to the data and parameters. We also provide the corresponding mechanical interpretations.

A modular extension of a finite element contact implementation

The structural mechanics module of the OOFEM finite element software has been developed to include various algorithms for solving contact mechanics problems. After developing small strain contact algorithms in the 2D domain and their extension to large strain applications [1, 2], the present contribution extends the existing framework to the 3D domain. After reviewing the current code and comparing existing solutions found in the literature, we identify the common ground between 2D and 3D applications, then propose and implement the necessary changes and additions to smoothly integrate the 3D support into existing code. Tests and example problems are presented to confirm the functionality of the resulting implementation.

Scalable preconditioning for the stabilized contact mechanics problem

We present a family of preconditioning strategies for the contact problem in fractured and faulted porous media. We combine low-order continuous finite elements to simulate the bulk deformation with piecewise constant Lagrange multipliers to impose the frictional contact constraints. This formulation is not uniformly inf-sup stable and requires stabilization. We improve previous work by Franceschini et al. (2020) by introducing a novel jump stabilization technique that requires only local geometrical and mechanical properties. We then design scalable preconditioning strategies that take advantage of the block structure of the Jacobian matrix using a physics-based partitioning of the unknowns by field type, namely displacement and Lagrange multipliers. The key to the success of the proposed preconditioners is a pseudo-Schur complement obtained by eliminating the Lagrange multiplier degrees of freedom, which can then be efficiently solved using an optimal multigrid method. Numerical results, including complex real-world problems, are presented to illustrate theoretical properties, scalability and robustness of the preconditioner. A comparison with other approaches available in the literature is also provided.

Bir Yarı-düzlemde Yüzeyaltı Termoelastik Temas Gerilmelerinin Sıcaklık Bağımlı Özellikler Kullanılarak Belirlenmesi

Contact mechanics problem between a rigid punch and a homogenous half-plane is examined considering frictional heat generation. Friction between the sliding rigid punch and the surface of the half-plane leads to a frictional heat which directly flows towards the half-plane material without a loss, and it changes the material’s thermoelastic properties. Calculation of subsurface stresses is crucial in the aspect of mechanical design of components since most of failures arise from fatigue and fracture at regions where subsurface stresses reach higher levels. In order to solve the problem, an iterative algorithm is developed based on the finite element method. Steady state subsurface contact stresses are obtained once the frictional heat on the contact surface reaches equilibrium. Subsurface stresses are calculated for different values of punch sliding velocity and coefficient of friction. It is observed that difference between subsurface contact stresses calculated based on temperature dependent and temperature independent properties is remarkable. Higher values of punch velocity and coefficient of friction leads to greater amount of heat generation, and percent difference between stresses reaches significant level especially near the contact surface. The utilization of temperature dependent material properties provides better approximation in assessing fatigue and fracture behavior of machine parts subjected to frictional contact with heat generation.

"The penalty method for generalized mixed variational-hemivariational inequality problems"

"It is well known that many popular variational, quasi-variational, hemivariational inequalities and variational inclusions involving constraints in a Banach space to convert a fixed point problems for finding the solution of such problems. This paper is to infuse a sequence of penalized problems without constraints and we show under the few reasonable assumptions to the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems is nonempty. As an application, we explore two complicated partial differential systems of elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints."

Sensitivity Analysis of a Scalar Mechanical Contact Problem with Perturbation of the Tresca’s Friction Law

This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca’s friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular, the friction threshold involved in the Tresca’s friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence, we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally, numerical simulations are provided in order to illustrate our main result.

From the Fan-KKM principle to extended real-valued equilibria and to variational-hemivariational inequalities with application to nonmonotone contact problems

This paper starts off by the celebrated Knaster–Kuratowski–Mazurkiewicz principle in the formulation by Ky Fan. We provide a novel variant of this principle and build an existence theory for extended real-valued equilibrium problems with general, then monotone and pseudomonotone bifunctions. We develop our existence theory first in general topological vector spaces, then in reflexive Banach spaces, where we investigate the issue of coerciveness for existence on unbounded sets. Thereafter we use the Clarke generalized differential calculus for locally Lipschitz functions and derive existence results for nonlinear variational-hemivariational inequalities and hemivariational quasivariational inequalities. As application, we treat a unilateral contact problem in solid mechanics with nonmonotone friction.

Sensitivity analysis of a Tresca-Type Problem leads to Signorini’s conditions

The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H1(Ω) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar, we show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.

A cut finite-element method for fracture and contact problems in large-deformation solid mechanics

Cut finite-element methods (CutFEMs) belong to the class of methods that allow boundaries/interfaces to cut through the elements, which avoids any meshing/remeshing problems. This is highly convenient from a practical point of view, especially when non-stationary interfaces are considered, e.g. phase boundaries in solids, as the interfaces can move independently of the mesh. There are many research directions related to CutFEM, one of which focuses on the equations of solid mechanics. Initially, the developments centred on linear elasticity and, in the previous publication by the authors, the method has been extended to large deformations and arbitrary constitutive relations, while the focus has been on phase boundaries in solids and on localised chemical reaction fronts in coupled mechanics–diffusion–reaction systems. In this paper, the method is further extended to more complex physics of the interfaces — fracture, i.e. separation of the interface into two surfaces in the current configuration, and contact between the separated surfaces. Several cases are considered — fracture with linear and non-linear traction separation, contact without and with adhesion. Each incremental generalisation of the approach contains a prior approach as a particular case, e.g. the phase boundary problem is a particular case of the fracture problem. The contact problem is treated in an unbiased way — the weak form is symmetric with respect to the choice of the contact surfaces for the integration. The weak forms are derived from the total energy functional. The proposed method has been tested computationally for the case of linear elements and passed the so-called patch tests and the convergence rate tests demonstrating the asymptotically optimal rates.

A finite volume penalty based segment-to-segment method for frictional contact problems

This paper presents a new contact boundary condition for finite volume simulations of frictional contact problems involving geometrical and material non-linearities. Deformation of bodies in contact is described by the updated Lagrangian form of the momentum equation which is discretised in space using the cell-centred finite volume method. The proposed contact boundary condition is based on the finite volume implementation of the penalty based segment-to-segment contact force calculation method in which normal contact pressure, governed by a penalty law, is integrated across the discretised contact surface, enforcing contact constraints in an integral manner. Such an approach, as opposed to the pointwise contact force calculation algorithm, allows for more accurate and more robust treatment of the contact area edge, simultaneous calculation of contact forces on both contact surfaces as well as smoother contact force during large sliding. The proposed numerical method is tested on challenging mechanical contact problems showing very good agreement with the available benchmark results.

Convergence results for a class of multivalued variational–hemivariational inequality

In the present paper, we focus on a class of multivalued variational–hemivariational inequalities with pseudomonotone operator and constraint set in Hilbert spaces. By employing the penalty method and the Moreau–Yosida approximation technique, we construct an approximating problem for the original multivalued variational–hemivariational inequality under consideration. The main result in the paper shows that every weak cluster of the solution sequence for the approximating problem is always a solution of the original problem. Moreover, based on the obtained weak convergence result, another two strong convergence results are obtained when the condition of pseudomonotonicity is reinforced. Finally, we illustrate the application of our abstract results in the study of a frictionless contact problem in mechanics with unilateral constraint.

An efficient model order reduction scheme for dynamic contact in linear elasticity

The paper proposes an approach for the efficient model order reduction of dynamic contact problems in linear elasticity. Instead of the augmented Lagrangian method that is widely used for mechanical contact problems, we prefer here the Linear Complementarity Programming (LCP) method as basic methodology. It has the advantage of resulting in the much smaller dual problem that is associated with the governing variational principle and that turns out to be beneficial for the model order reduction. Since the shape of the contact zone depends strongly on the acting outer forces, the LCP for the Lagrange multipliers has to be solved in each time step. The model order reduction scheme, on the other hand, is applied to the large linear system for the displacements and computed in advance by means of an Arnoldi process. In terms of computational effort the reduction scheme is very appealing because the contact constraints are fully satisfied while the reduction acts only on the displacements. As an extension of our approach, we furthermore take up the idea of the Craig-Bampton method in order to distinguish between interior nodes and the nodes in the contact zone. A careful performance analysis closes the paper.

Artificial Neural Network Architecture for Prediction of Contact Mechanical Response

Predicting the contact mechanical response for various types of surfaces is and has long been a subject, where many researchers have made valuable contributions. This is because the surface topography has a tremendous impact on the tribological performance of many applications. The contact mechanics problem can be solved in many ways, with less accurate but fast asperity-based models on one end to highly accurate but not as fast rigorous numerical methods on the other. A mathematical model as fast as an asperity-based, yet as accurate as a rigorous numerical method is, of course, preferred. Artificial neural network (ANN)–based models are fast and can be trained to interpret how in- and output of processes are correlated. Herein, 1,536 surface topographies are generated with different properties, corresponding to three height probability density and two power spectrum functions, for which, the areal roughness parameters are calculated. A numerical contact mechanics approach was employed to obtain the response for each of the 1,536 surface topographies, and this was done using four different values of the hardness per surface and for a range of loads. From the results, 14 in situ areal roughness parameters and six contact mechanical parameters were calculated. The load, the hardness, and the areal roughness parameters for the original surfaces were assembled as input to a training set, and the in situ areal roughness parameters and the contact mechanical parameters were used as output. A suitable architecture for the ANN was developed and the training set was used to optimize its parameters. The prediction accuracy of the ANN was validated on a test set containing specimens not seen during training. The result is a quickly executing ANN, that given a surface topography represented by areal roughness parameters, can predict the contact mechanical response with reasonable accuracy. The most important contact mechanical parameters, that is, the real area of contact, the average interfacial separation, and the contact stiffness can in fact be predicted with high accuracy. As the model is only trained on six different combinations of height probability density and power spectrum functions, one can say that an output should only be trusted if the input surface can be represented with one of these.

Green's function method for dynamic contact calculations.

Resolving atomic scale details while capturing long-range elastic deformation is the principal difficulty when solving contact mechanics problems with computer simulations. Fully atomistic simulations must consider large blocks of atoms to support long-wavelength deformation modes, meaning that most atoms are far removed from the region of interest. Building on earlier methods that used elastic surface Green's functions to compute static substrate deformation, we present a numerically efficient dynamic Green's function technique to treat realistic, time-evolving, elastic solids. Our method solves substrate dynamics in reciprocal space and utilizes precomputed Green's functions that exactly reproduce elastic interactions without retaining the atomic degrees of freedom in the bulk. We invoke physical insights to determine the necessary number of explicit substrate layers required to capture the attenuation of subsurface waves as a function of surface wave vector. We observe that truncating substrate dynamics at depths that fall as a power of wave vector allows us to accurately model wave propagation without implementing arbitrary damping. The framework we have developed substantially accelerates molecular dynamics simulations of large elastic substrates. We apply the method to single asperity contact, impact, and sliding friction problems and present our preliminary findings.