Non-penetration Constraint Research Articles

Mixed-formulation with non-penetration constraint for planar composite beams in partial interaction

Mixed-formulation with non-penetration constraint for planar composite beams in partial interaction

A smoothed assumed enhanced strain method for frictional contact with constant strain elements

This paper presents a finite element framework for imposing frictional contact conditions on embedded fracture faces, implemented by the constant-strain assumed enhanced strain (AES) method, where penalty method is used to impose both non-penetration constraint and Coulomb's law of friction. The proposed constant-strain AES method for modeling embedded frictional contact can be cast into an integration algorithm similar to those used in the classical plasticity theory, where displacement jump is calculated from the local traction equilibrium at Gauss point, so the method does not introduce any additional global degrees of freedom. Moreover, constant-strain elements are often desirable in practice because they can be easily created automatically for large-scale engineering applications with complicated geometries. As encountered in other enriched finite element methods for frictional contact, the problem of normal contact pressure oscillations is also observed in the constant-strain AES method. Therefore, we developed a strain-smoothing procedure to effectively mitigate the oscillations. We investigated and verified the proposed AES framework through several numerical examples, and illustrated the capability of this method in solving challenging nonlinear frictional contact problems.

A Mortar Finite Element Formulation for Large Deformation Lubricated Contact Problems with Smooth Transition Between Mixed, Elasto-Hydrodynamic and Full Hydrodynamic Lubrication

This work proposes a novel model and numerical formulation for lubricated contact problems describing the mutual interaction between two deformable 3D solid bodies and an interposed fluid film. The solid bodies are consistently described based on nonlinear continuum mechanics allowing for finite deformations and arbitrary constitutive laws. The fluid film is modelled as a quasi-2D flow problem governed by the (thickness-)averaged Reynolds equation. In contrast to existing approaches, the proposed model accounts for the co-existence of frictional contact tractions and hydrodynamic fluid tractions at every local point on the contact surface of the interacting bodies and covers the entire range of lubrication in one unified modelling framework with smooth transition between these different regimes. From a physical point of view, this approach can be considered as a model for the elastic deformation of asperities on the lubricated contact surfaces. The finite element method is applied for spatial discretization of the 3D solid-mechanical problems and the 2D interface effects, consisting of the averaged Reynolds equation governing the fluid film and the non-penetration constraint of the mechanical contact problem. A consistent and accurate model behavior is demonstrated by studying several challenging benchmark test cases.

A barrier method for frictional contact on embedded interfaces

We present a barrier method for treating frictional contact on interfaces embedded in finite elements. The barrier treatment has several attractive features, including: (i) it does not introduce any additional degrees of freedom or iterative steps, (ii) it is free of inter-penetration, (iii) it avoids an ill-conditioned matrix system, and (iv) it allows one to control the solution accuracy directly. We derive the contact pressure from a smooth barrier energy function that is designed to satisfy the non-penetration constraint. Likewise, we make use of a smoothed friction law in which the stick–slip transition is described by a continuous function of the slip displacement. We discretize the formulation using the extended finite element method to embed interfaces inside elements, and devise an averaged surface integration scheme that effectively provides stable solutions without traction oscillations. Subsequently, we develop a way to tailor the parameters of the barrier method to embedded interfaces, such that the method can be used without parameter tuning. We verify and investigate the proposed method through numerical examples with varied levels of complexity. The numerical results demonstrate that the proposed method is remarkably robust for challenging frictional contact problems, while requiring low cost comparable to that of the penalty method.

Bayesian inversion for steady flow in fractured porous media with contact on fractures and hydro-mechanical coupling

The paper is motivated by a strong interest in numerical analysis of flow in fractured porous media, e.g., rocks in geo-engineering applications. It follows the conception of porous media as a continuum with fractures which are represented as lower dimensional objects. In the paper, the finite element discretization of the flow in coupled continuum and fractures is used; fluid pressures serve as the basic unknowns. In many applications, the flow is connected with deformations of the porous matrix; therefore, the hydro-mechanical coupling is also considered. The fluid pressure is transferred to the mechanical load in both pores and fractures and the considered mechanical model involves elastic deformations of the porous matrix and opening/closing of the fractures with the non-penetration constraint. The mechanical model with this constraint is implemented via the technique of the Lagrange multipliers, duality formulation, and combination with a suitable domain decomposition method. There is usually lack of information about problem parameters and they undergo many uncertainties coming e.g. from the heterogeneity of rock formations and complicated realization of experiments for parameter identification. These experiments rarely provide some of the asked parameters directly but require solving inverse problems. The stochastic (Bayesian) inversion is natural due to the mentioned uncertainties. In this paper, the implementation of the Bayesian inversion is realized via Metropolis-Hastings Markov chain Monte Carlo approach. For the reduction of computational demands, the sampling procedure uses the delayed acceptance of samples based on a surrogate model which is constructed during a preliminary sampling process. The developed hydro-mechanical model and the implemented Bayesian inversion are tested on two types of model inverse problems.

A large deformation hybrid isogeometric-finite element method applied to cohesive interface contact/debonding

A hybrid NURBS-based isogeometric-finite element discretization technique is presented to take advantages of IGA for problems with regions demanding higher geometric accuracy or interpolation order. The validity of this coupled discretization is tested through standard patch test. Also, for demonstrative purposes, the proposed methodology is then applied to 2D interface contact/debonding of fiber–matrix as well as a 2D double cantilever beam peeling problem. Adopting a novel approach, the mortar method is used to account for both cohesive behavior and non-penetration constraint in the interface. Non-penetration constraint is enforced via Lagrange multiplier technique along with a mixed-mode cohesive law. The validity of the proposed unified contact/debonding formulation is tested against a contact/tension patch test. The stress contours after simulations show a smooth variation across IG and FE domains further demonstrating the coupling’s eligibility. Moreover, smooth contact pressure distributions are obtained along the interfaces resulting from higher-order NURBS basis functions. Having significantly less DOFs, the hybrid IG–FE discretization also behaved more robustly compared with linear FEs.

Dynamic modeling method of flexible bodies with contact/impact based on interactive mode

To solve the contact/impact problem of flexible bodies in arbitrary shape, the finite element method is widely used to discretize the contact bodies. In the finite element method, two contact models are mainly used to compute the contact force, i.e., penalty function method and additional constraint method, which are different in constraint imposition strategy. The penalty function method regards the contact effect as a force function of local penetration at the contact point and its rate. This method has gained significant popularity because it does not bring extra dimensions to the dynamic equations and does not need to solve constraint equations either. However, as the non-penetration constraint is not precisely satisfied in the contact process when using the penalty function method, the accuracy of the numerical simulation depends on the choice of the penalty parameter. On the other hand, the additional constraint method can strictly satisfy the contact constraint condition by introducing the Lagrange multipliers into the dynamic equations, but the method poses some numerical difficulties due to the additional effort required to solve the multipliers. Considering the advantages and disadvantages of the two contact methods, the interactive mode method is proposed. This method divides the whole model into local static module and main dynamics module. The static module establishes a local finite element model of the contact region to compute the contact force, and the main dynamics module is used to obtain the kinematic variables of the whole body. In the simulation, the two modules are coupled by exchanging displacements and forces in each time step. In the current integration step, the main dynamics module provides the displacements of the boundaries of the local contact region at first, the values are transferred to the local finite model to compute the contact force next, and then the contact force is fed back to the dynamics module for the calculation of the next step. The proposed method combines the advantages of both the additional constraint method and the penalty function method, in which not only the artificial selection of penalty parameter is avoided, but also the non-penetration constraint of local contact region is satisfied and the numerical solution is convenient. The validity of the proposed method is verified by the comparison between simulation results and experimental results of a rod-plate impact case. Furthermore, a multi-point impact problem of a slider sliding in the gap chute is presented to validate the proposed method of dealing with the general impact problem.

A Geometrically Exact Contact Model for Polytopes in Multirigid-Body Simulation

We present a formulation of nonpenetration constraint between pairs of polytopes which accounts for all possible combinations of active contact between geometric features. This is the first formulation that exactly models the body geometries near points of potential contact, preventing interpenetration while not overconstraining body motions. Unlike many popular methods, ours does not wait for penetrations to occur as a way to identify which contact constraints to enforce. Nor do we overconstrain by representing the free space between pairs of bodies as convex, when it is in fact nonconvex. Instead, each contact constraint incorporates all feasible potential contacts in a way that represents the true geometry of the bodies. This ensures penetration-free, physically correct configurations at the end of each time step while allowing bodies to accurately traverse the free space surrounding other bodies. The new formulation improves accuracy, dramatically reduces the need for ad hoc corrections of constraint violations, and avoids many of the inevitable instabilities consequent of other contact models. Although the dynamics problem at each time step is larger, the inherent stability of our method means that much larger time steps can be taken without loss of physical fidelity. As will be seen, the results obtained with our method demonstrate the effective elimination of interpenetration, and as a result, correction-induced instabilities, in multibody simulations.

Discussion of the Gear–Gupta–Leimkuhler method for impacting mechanical systems

In multibody simulation, the Gear-Gupta-Leimkuhler method for only persistent contacts enforces constraints on position and velocity level at the same time. It yields a robust numerical discretization of differential algebraic equations avoiding the drift-off effect. In this work, we carry over these benefits to impacting mechanical systems with unilateral constraints. For this kind of a mechanical system, adding the position level constraint to a timestepping scheme on velocity level even maintains physical consistency of the impulsive discretization. Hence, we propose a timestepping scheme based on Moreau's midpoint rule which enables to achieve not only compliance of the impact law but also of the non-penetration constraint. The choice of a decoupled and consecutive evaluation of the respective constraints can be interpreted as a not energy-consistent projection to the non-penetration constraint at the end of each time step. It is the implicit coupling of position and velocity level which yields satisfactory results. An implicit evaluation of the right hand side improves stability properties without additional cost. With the prox function formulation, the overall set of nonsmooth equations is solved by a Newton scheme. Results from simulations of a slider-crank mechanism with unilateral constraints demonstrate the capability of our approach.

Configuration-Based Optimization for Six Degree-of-Freedom Haptic Rendering for Fine Manipulation

Six-degree-of-freedom (6-DOF) haptic rendering for fine manipulation in narrow space is a challenging topic because of frequent constraint changes caused by small tool movement and the requirement to preserve the feel of fine-features of objects. In this paper, we introduce a configuration-based constrained optimization method for solving this rendering problem. We represent an object using a hierarchy of spheres, i.e., a sphere tree, which allows faster detection of multiple contacts/collisions among objects than polygonal mesh and facilitates contact constraint formulation. Given a moving graphic tool as the avatar of the haptic tool in the virtual environment, we compute its quasi-static motion by solving a configuration-based optimization. The constraints in the 6D configuration space of the graphic tool is obtained and updated through online mapping of the nonpenetration constraint between the spheres of the graphic tool and those of the other objects in the three-dimensional physical space, based on the result of collision detection. This problem is further modeled as a quadratic programming optimization and solved by the classic active-set methods. Our algorithm has been implemented and interfaced with a 6-DOF Phantom Premium 3.0. We demonstrate its performance in several benchmarks involving complex, multiregion contacts. The experimental results show both the high efficiency and stability of haptic rendering by our method for complex scenarios. Nonpenetration between the graphic tool and the object is maintained under frequent contact switches. Update rate of the simulation loop including optimization and constraint identification is maintained at about 1 kHz.

Equilibration techniques for solving contact problems with Coulomb friction

In this paper, we consider residual and equilibrated error indicators for contact problems with Coulomb friction. The contact problem is handled within the abstract framework of saddle point problems. More precisely, the non-penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual Lagrange multiplier space. Thus from the displacement, we can easily compute in a local post-process the Lagrange multiplier which acts as a Neumann condition on the possible contact zone. Having computed the discrete Lagrange multiplier, we can apply standard error estimators by replacing the unknown Neumann data by its approximation. As it is shown in [1], this results in an error estimator for a one-sided contact problem without friction. Here, we consider more general situations and discuss two additional contact terms which measure the non-conformity of the discrete Lagrange multiplier. Numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process.

Modeling compliant non-penetration constraint for VP motion simulation

Modeling compliant non-penetration constraint for VP motion simulation

A Numerical–Analytical Solution for Dynamics of Composite Delaminated Beam with Piezoelectric Actuator, with Account of Nonpenetration Constraint for the Delamination Crack Faces

In this work, a new approach is developed for the dynamic analysis of a composite beam with an interply crack, in which a physically impossible interpenetration of the crack faces is prevented by imposing a special constraint, leading to nonlinearity of the formulated boundary value problem and taking account of a contact interaction of the crack faces. A variational formulation of the problem and partial differential equations of motion with boundary conditions are developed, and solutions of example problems for a piezo-actuated cantilever beam are presented in the form of a series in terms of eigenfunctions of the associated nonself-adjoint eigenvalue problem. A noticeable difference of forced vibrations of the delaminated and undelaminated beams due to the contact interaction of the crack faces is predicted by the developed model.

Maximum Stiffness Design for Elastic Bodies in Contact

An optimality criteria approach for the stiffness design problem is developed for the case of two elastic bodies in frictionless contact. The contact problem is analyzed by appending the nonpenetration constraint to the system potential energy. Necessary and sufficient conditions for stiffness maximization are derived. A computational method for design problem solution is developed and numerical results for a beam on a foundation are displayed.