Unilateral Constraints Research Articles

Stability analysis of periodic solutions computed for blade-tip/casing contact problems

This article focuses on the stability assessment of mechanical systems featuring a nonlinear interface on which are applied unilateral contact constraints. The regularized-Lanczos harmonic balance method is used to compute periodic solutions for each system. For each considered mechanical system, the implementation of distinct stability assessment numerical algorithms (including the Newmark 2 n -pass method and Hill’s method using eigenvalue sorting) relying on Floquet theory allows for an in-depth investigation of their convergence and accuracy. From a single-degree of freedom impactor impacting a fixed obstacle to the industrial finite element model of a transonic compressor blade impacting a deformed casing, the gradually increasing complexity of the considered mechanical systems provides new insight on which stability assessment numerical algorithm may be the best suited when dealing with contact nonlinearities. From a numerical standpoint, attention is paid to specific developments that were identified as key for enhancing the numerical efficiency of the stability assessment process including the iterative solver, the computation of analytical derivatives, the scaling of the problem and the use of fast Fourier transforms. Particular attention is also paid to the influence of the Lanczos filtering procedure and its implications in terms of stability assessment. While it is found that both the Newmark 2 n -pass method and Hill’s method with eigenvalue sorting can be successfully applied on large industrial systems featuring a nonlinear interface, the faster convergence of Hill’s method is underlined and suggests it may be a better option for an accurate and efficient stability assessment on which Lanczos filtering has a negligible influence. • Local stability analysis of periodic solutions computed using a Harmonic Balance Method. (HBM) based numerical strategy for contact problems of increasing severity. • Analytical expression of blade-tip/casing contact forces related Jacobian matrices. • Numerical considerations for efficient HBM computations and in-depth comparison of time and frequency domain-based stability assessment algorithms. • Detailed application on both academical impactors and an industrial blade: NASA rotor 37. • Influence of Lanczos filtering over stability assessment.

Formation tracking control design for uncertain artificial swarm systems under inequality constraints: Leakage and dead-zone type

The paper proposes a formation tracking control method for the uncertain artificial swarm systems under the inequality constraints. Not only can the agents perform swarm behaviors (e.g., convergence, formation and avoidance of collision), but they can also track the fixed targets in a constrained area (which is formulated as the inequality constraints, such as unilateral constraint and bilateral constraint.). The swarm behaviors are creatively considered as the servo constraints or the control objectives for the swarm agents. Based on the Udwadia–Kalaba (U–K) equation, those prescribed behaviors are realized by a model-based control design (that is the servo constraints force model-based feedforward control). To deal with the inequality constraints in the formation tracking process, a differential homeomorphism transformation is used to relieve the environmental constraints for the swarm agents. Moreover, the uncertainty of the swarm agents (i.e., the parameter uncertainty in modeling and the external disturbances) is considered, which is time-varying and unknown (but bounded). An uncertainty estimation method with dead-zone and leakage term is designed to calculate the possible upper bound of the uncertainty. In virtue of the estimated upper bound of the uncertainty, a robust control is designed for the uncertain swarm agents to obey the prescribed swarm behaviors in the formation tracking task. The system performances of the artificial swarm systems under the proposed control are theoretically guaranteed by a range of rigorous theorems and numerically verified by the simulations of three agents.

On normal compliance quasistatic viscoplastic contact problem with friction, damage and foundation memory property

The goal of this paper is to study a mathematical model which describes the quasistatic frictional contact between aviscoplastic body and a foundation. The contact is modeled with normal compliance, unilateral constraint, friction, damage andmemory term, which also includes foundation memory property. We present the classical formulation of the problem and derivevariational-hemivariational formulation. Finally we provide a unique weak solution of the problem.

Buckling versus unilateral constraint for a multistable metamaterial element.

A structural element is designed and investigated, forming the basis for the development of an elastic multistable metamaterial. The leitmotif of the structural design is the implementation of a strut characterized by a bifurcation occurring at either vanishing tensile or compressive load. It is shown that buckling at null load leads to a mechanical equivalence with a unilateral constraint formulation, introducing shocks in dynamics. Towards a future analysis of the latter, the nonlinear quasi-static response is investigated, showing the multistable character of the structure, which may appear as bistable or tetrastable. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

A new class of history-dependent quasi variational–hemivariational inequalities with constraints

In this paper we consider an abstract class of time-dependent quasi variational–hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational–hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

Representing and Computing the B-Derivative of the Piecewise-Differentiable Flow of a Class of Nonsmooth Vector Fields

Abstract This paper concerns first-order approximation of the piecewise-differentiable flow generated by a class of nonsmooth vector fields. Specifically, we represent and compute the Bouligand (or B-)derivative of the piecewise-differentiable flow generated by a vector field with event-selected discontinuities. Our results are remarkably efficient: although there are factorially many “pieces” of the derivative, we provide an algorithm that evaluates its action on a tangent vector using polynomial time and space, and verify the algorithm's correctness by deriving a representation for the B-derivative that requires “only” exponential time and space to construct. We apply our methods in two classes of illustrative examples: piecewise-constant vector fields and mechanical systems subject to unilateral constraints.

Axisymmetric Contact Problems for Composite Pressure Vessels

The present analysis is conducted for the evaluation of contact pressures of axisymmetric shells made of laminated composites or functionally graded materials. This class of problems is usually called the Signorini–Fichera problem (unilateral constraints) and can be solved as the lower-bound problem. The numerical solution of this problem is proposed both for symmetric and unsymmetric shell configurations. The first-ply-failure of such structures is considered. It is demonstrated that the failure occurs at the end of the contact area corresponding to the appearance of stress concentration of radial concentrated forces.

State observers for the time discretization of a class of impulsive mechanical systems

AbstractIn this work, we investigate the state observer problem for linear mechanical systems with a single unilateral constraint, for which neither the impact time instants nor the contact distance is explicitly measured. We propose to attack the observer problem by transforming and approximating the original continuous‐time system by a discrete linear complementarity system (LCS) through the use of the Schatzman–Paoli scheme. From there, we derive a deadbeat observer in the form of a linear complementarity problem. Sufficient conditions guaranteeing the uniqueness of its solution then serve as observability conditions. In addition, the discrete adaptation of an existing passivity‐based observer design for LCSs can be applied. A key point in using a time discretization is that the discretization acts as a regularization, that is, the impacts take place over multiple time steps (here two time steps). This makes it possible to render the estimation error dynamics asymptotically stable. Furthermore, the so‐called peaking phenomenon appears as singularity within the time discretization approach, posing a challenge for robust observer design.

On Harmonic Balance Method-based Lagrangian contact formulations for vibro-impact problems

This article assesses two Harmonic Balance Method-based numerical Lagrangian strategies for the characterization of the periodic response of a mechanical system subject to unilateral contact constraints. The model used for evaluation is an academic rod model facing an obstacle that is firstly considered as equivalent to a stiffness (flexible) and then as a rigid obstacle. Nonlinear frequency response curves are obtained through an arc-length continuation procedure and confronted to the corresponding time integration simulations. An in-depth comparative analysis is conducted on the time signals obtained through both frequency domain strategies: (1) linear complementarity problem (LCP-HBM) and (2) dynamic Lagrangian frequency time (DLFT-HBM). The unilateral contact conditions are thoroughly investigated in the time domain and compared with the reference time integration simulations. Particular attention is paid to the relation between both methods and it is shown that the DLFT-HBM is asymptotically equivalent to the LCP-HBM. This study also shows that the oversampling of time signals inside alternating frequency–time procedures generates non-physical high harmonics. Finally, a convergence analysis is conducted in order to assess the influence of the different numerical parameters on the respect of the unilateral contact laws.

Dynamic manipulability measure for on-orbit manipulation by frictional contacts

Due to unilateral constraints of frictional contacts and the dynamic coupling phenomenon, the study on the unfixed manipulation through frictional contacts for multi-arm space robots is few carried out. This paper proposes a new measure method to assess the dynamic manipulability of the unfixed manipulation through frictional contacts. In particular, the explicit map from the base actuation and joint torques to the object acceleration, the base acceleration, and the internal force is formulated under unilateral constraints of frictional contacts. Considering typical control scenarios of both free-floating and free-flying systems, dynamic manipulabilities are derived separately. As an illustrative example, the dynamic manipulability of a planar dual-arm space robot is computed and reported, which visualizes the influence of different system parameters.

Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

A system of a first order history-dependent evolutionary variational– hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem for history dependent operators, results on the well-posedness of the system are proved. Existence, uniqueness, continuous dependence of the solution on the data, and the solution regularity are established. Two applications of dynamic problems from contact mechanics illustrate the abstract results. First application is a unilateral viscoplastic frictionless contact problem which leads to a hemivariational inequality for the velocity field, and the second one deals with a viscoelastic frictional contact problem which is described by a variational inequality.

Fast and reliable probabilistic face embeddings based on constrained data uncertainty estimation

Probabilistic Face Embeddings (PFE) can improve face recognition performance in unconstrained scenarios by integrating data uncertainty into the feature representation. However, the matching speed of PFE is too slow to be applied to large-scale face recognition or retrieval applications. Moreover, since deep learning-based data uncertainty estimation tends to be over-confident, the recognition performance of PFE is unstable. This paper proposes a probabilistic face embedding method to improve the robustness and speed of PFE. Specifically, the mutual likelihood score (MLS) metric used in PFE is simplified by using a one-dimensional variance to approximate the data uncertainty of face feature, to speed up the matching of probabilistic embedding pairs. Then, a unilateral constraint loss is proposed to limit the variation range of the lower part of the estimated data uncertainties, which can solve the problem of accuracy degradation in high-quality images. In addition, a feature fusion method based on temperature scaling is proposed, which can adjust the fusion weights of different quality images to improve the performance of video face recognition. Comprehensive experiments show that the proposed method can achieve comparable or better results in 6 benchmarks than the state-of-the-art methods with a less computational cost at the matching process. The code of our work is publicly available in GitHub (https://github.com/KaenChan/ProbFace).

"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."

Dynamic viscoelastic unilateral constrained contact problems with thermal effects

A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations.

Optimal Control Problems for Set-Valued Quasivariational Inequalities with Applications

In this paper we investigate the optimal control problem for set-valued quasivariational inequality with unilateral constraints. Under suitable conditions, we prove that the solution to the current optimal control problem converges to a solution to old control problems. By way of application, we utilize our results presented in the paper to study the optimal control associated with boundary value problems which is described by frictional contact problems and a stationary heat transfer problem with unilateral constraints.

Dynamics of a viscoelastic membrane with gradient constraint

Taking into account inertial and viscosity effects, we consider the dynamics of a two dimensional membrane subjected to an unilateral constraint on its deformation gradient. Specifically, due to the constitutive law, we assume that higher deformations lock the material, leading to the inequality |∇u|≤g, where u denotes the displacement of the membrane and g is a certain positive threshold. We then introduce the concepts of weak and generalised solutions to the associated wave equation, and prove the existence of them for rather general data and homogeneous Dirichlet boundary conditions. The presence of the gradient constraint provides the existence of a Lagrange multiplier λ related to the existence of a reaction term ϒ, which corresponds to a strongly nonlinear term in the wave equation. We then extend the existence result to a weak form of the Neumann type boundary condition αu+∂u∂ν+∂u˙∂ν+ϒ⋅ν=0, for any α≥0, and we show that these solutions tend, as α→∞, in a certain sense to a solution of the homogeneous Dirichlet constrained problem.

A unilateral constraint contact problem for viscoelastic materials with wear

The aim of this work is to investigate a mathematical model that describes the frictional contact between viscoelastic materials with a long memory and a moving foundation. The contact condition is modeled with a version of normal compliance condition with unilateral constraint in which wear of foundation is considered. Friction contact condition follows a sliding version of Coulomb's law of dry friction. We derive a variational formulation of the problem, which couples a variational inequality and an integral equation. Then, we obtain the unique weak solvability of the contact problem by virtue of a fixed point theorem. By using the penalization method, we discuss the convergence of the contact problem. Further, we obtain the unique solution of the penalized problem and this solution converges to the solution of the original contact model as the penalization parameter converges to zero.

Double phase Dirichlet problems with unilateral constraints

We consider a double phase Dirichlet problem with both convex and nonconvex unilateral constraints (variational-hemivariational inequality). Using variational techniques and tools from nonsmooth analysis (convex and nonconvex), we establish the existence of a nontrivial bounded solution.

Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints

We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

The Optimal Control Problems for Generalized Elliptic Quasivariational Inequalities

In this article, we propose an optimal control problem for generalized elliptic quasi-variational inequality with unilateral constraints. Then, we discuss the sufficient assumptions that ensure the convergence of the solutions to the optimal control problem. The proofs depend on convergence results for generalized elliptic quasi-variational inequalities, obtained by the arguments of compactness, lower semi-continuity, monotonicity, penalty and different estimates. As an application, we addressed the abstract convergence results in the analysis of optimal control associated with boundary value problems.