Unilateral Constraints Research Articles

Nonnegative Boundary Control of 1D Linear Heat Equations

We consider the controllability of a one dimensional heat equation with nonnegative boundary controls. Despite the controllability in any positive time of this system, the unilateral nonnegativity control constraint causes a positive minimal controllability time. In this article, it is proved that at the minimal time, there exists a nonnegative control in the space of Radon measures, which consists of a countable sum of Dirac impulses.

An explicit time-integrator with singular mass for non-smooth dynamics

This article addresses the simulation of non-smooth dynamics problems with unilateral contact constraints between rigid and deformable bodies. It proposes a modified CD-Lagrange scheme with a singular mass matrix. This scheme is explicit, and based on a contact condition on velocity. The formulation is designed for a 1D impact problem between deformable and rigid body with unilateral constraint. The singular mass matrix allows to get a more accurate energy balance on the discrete system, especially during non-smooth events. An extension is presented then for the 3D cases. Its implementation is easy, and fully compatible with large deformations or non-linear materials. Indeed it consists only in adding a numerical parameter for each contact node. The energy balance for the singular 3D formulation is improved compared to the consistent one.

Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture

Gradient damage models used in phase-field approaches to brittle fracture are characterised by material softening and instabilities. We present novel numerical techniques for the bifurcation and stability analysis along quasi-static evolution paths as well as practical tools to select stable evolutions. Our approach stems from the variational approach to fracture and the theory of rate-independent irreversible processes whereby a quasi-static evolution is formulated in terms of incremental energy minimisation under unilateral constraints. Focusing on the discrete setting obtained with finite elements techniques, we discuss the links between bifurcation criteria for an evolution and stability of equilibrium states. Key concepts are presented through the analytical solution of a two-degrees-of-freedom model featuring a continuum family of bifurcation branches. We introduce numerical methods to (i) assess (second-order) stability conditions for time-discrete evolutions subject to damage irreversibility, and (ii) to select possible stable evolutions based on an energetic criterion. Our approach is based on the solution of a coupled eigenvalue problem which accounts for the time-discrete irreversibility constraint on damage. Several numerical examples illustrate that this approach allows us to filter out unstable solutions provided by standard (first-order) minimisation algorithms as well as to effectively compute stable evolution paths. We demonstrate our purpose on a multifissuration problem featuring complex fracture patterns, to show how the eigenvalue analysis enables to compute and retrieve morphological properties of emerging cracks.

Generalized penalty method for history-dependent variational–hemivariational inequalities

We consider a history-dependent variational–hemivariational inequality with unilateral constraints in a reflexive Banach space. The unique solvability of the inequality follows from an existence and uniqueness result obtained in Sofonea and Migórski (2016, 2018). In this current paper we introduce and study a generalized penalty method associated to the inequality. To this end we consider a sequence of generalized penalty problems, governed by a parameter λn and an operator Pn. We prove the unique solvability of the penalty problems as well as the convergence of corresponding solutions sequence to the solution of original problem. These results extend the previous results in Sofonea et al. (2018) and Xiao and Sofonea (2019). Finally, we illustrate them in the study of a history-dependent problem with unilateral boundary conditions which describes the quasistatic evolution of a rod–spring system under the action of given applied force.

A lightweight network with bidirectional constraints for single image super-resolution

The existing methods based on the convolutional neural network (CNN) show excellent performance in single image super-resolution (SISR). Massive convolutional layers can enhance these CNN-based methods, but they increase memory consumption due to a large number of parameters. In addition, the unilateral constraint on SISR only focuses on the up-sampling process and ignores image degradation, limiting the accuracy of convergence in network training. In response to these problems, we propose a lightweight network with bidirectional constraints (LNBC) for SISR. We present an extended layer named enhanced cycle residual block (CRB), then develop a lightweight network with CRB as the feature inference layer. CRB improves feature expression ability by alternating and multiplexing convolutional layers without increasing parameters. In addition, unlike existing methods that only constrain the up-sampling process, we propose bidirectional constraints on SISR named cycle consistency verification (CCV). In CCV, the introduced degradation network simulates image degradation, and provides constraints on image degradation for the up-sampling network in joint training. Image up-sampling and degradation constitute bidirectional constraints to tighten the convergence of the up-sampling network. Experiments show that LNBC advances comparison methods in the compromise of image super-resolution performance, memory overhead and running time.

English

We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D\subset\R^d$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $\cP$. We associate to Problem $\cP$ an optimal control problem, denoted by $\cQ$. Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\wK$, we provide results concerning the well-posedness of problems $\cP$ and $\cQ$. Our main results are Theorems 14 and 18, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem $\cP$, constructed with particular choices of $G$ and $\wK$. We prove that Theorems 14 and 18 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results.

The harmonic balance method with arc-length continuation in blade-tip/casing contact problems

This article presents a Harmonic Balance Method-based numerical strategy to provide a qualitative numerical characterization of a compressor blade’s dynamics when structural contacts occur with the surrounding casing. The mitigation of the Gibbs phenomenon follows a two-pronged approach: (1) a regularization of the unilateral contact law and (2) the adjustment of the Fourier coefficients by means of a Lanczos filtering technique. In order to validate the proposed approach, it is first applied to an academic cantilever rod undergoing unilateral contact constraints. An in-depth comparative analysis of the obtained results—with an emphasis on displacements, contact forces and velocities—with respect to a time integration-based reference numerical strategy underlines the relevance and accuracy of the proposed methodology. The latter is then applied to the vibration analysis of an industrial compressor blade: NASA rotor 37. For a given contact configuration, obtained results are thoroughly compared to those obtained with a previously published time integration-based numerical strategy with a distinct contact treatment algorithm. Particular attention is paid to demonstrate the accuracy of the methodology for the prediction of displacements, contact forces, velocities as well as stress fields within the blade. Notably, it is evidenced that the proposed methodology, contrary to the reference time integration-based numerical strategy, is able to capture the exact location of the blade’s nonlinear resonance.

On the use of multibody dynamics techniques to simulate fluid dynamics and fluid–solid interaction problems

A multibody dynamics-based solution to the fluid dynamics problem is compared herein to two established Lagrangian-based techniques used by the computational fluid dynamics (CFD) community. The multibody dynamics-based solution has two salient attributes: it enforces the incompressibility condition through bilateral kinematic constraints, and it treats the coupling with the solid phase via unilateral kinematic constraints. The multibody dynamics-based solution, called herein the Kinematically Constrained Smoothed Particle Hydrodynamics (KCSPH) method, is a Lagrangian approach to solving the CFD problem. It relies on the Smoothed Particle Hydrodynamics (SPH) method to discretize the spatial differential operators in the Navier–Stokes equations, and on the Newton–Euler equations of multibody dynamics to convect the SPH particles forward in time. We show that the multibody dynamics-based approach is efficient and accurate by comparing its performance with the two most commonly used SPH algorithms in the CFD community: the weakly compressible SPH (WCSPH), and the implicit SPH (ISPH) methods. The comparison is carried out in conjunction with four tests: an incompressibility benchmark test, dam break, floating cylinder, and sloshing tank. We conclude that KCSPH is a robust alternative to conventional CFD approaches for fluid–solid interaction (FSI) problems with complex/moving boundaries. The solvers and models used herein are publicly available in an open-source software called Chrono; the implementations use GPU (for WCSPH and ISPH), and multicore CPU (for KCSPH) parallel computing.

The inverse-deformation approach to fracture

We propose a one-dimensional, nonconvex elastic constitutive model with higher gradients that can predict spontaneous fracture at a critical load via a bifurcation analysis. It overcomes the problem of discontinuous deformations without additional field variables, such as damage or phase-field variables, and without a priori specified surface energy. Our main tool is the inverse deformation, which can be extended to be a piecewise smooth mapping even when the original deformation has discontinuities describing cracks opening. We exploit this via the inverse-deformation formulation of finite elasticity due to Shield and Carlson, including higher gradients in the energy. The problem is amenable to a rigorous global bifurcation analysis in the presence of a unilateral constraint. Fracture under hard loading occurs on a bifurcating solution branch at a critical applied stretch level, with a sudden drop to zero stress. Fractured solutions are found to have sharply defined cracks with surface energy arising from higher gradient effects.

Single image Dehazing algorithm based on double exponential attenuation model

Single image dehazing is a challenging task because haze seriously affect the visibility of images, and it further brings inconvenience to advanced computer vision analysis. In this paper, a novel and efficient algorithm is proposed to remove haze from a single image. Instead of obtaining transmission from fixed priors and constraints, a double exponential attenuation model (DEA model) based on color attenuation prior is established to compute transmission adaptively. DEA model contains two single exponential attenuation parts that can compensate each other. In addition, a normal unilateral constraint rule (NUCR) is designed to make the transmission more accurate. More importantly, DEA model has a dynamic adjustment function with different weights for different inputs. To circumvent distortion in sky and bright areas, an effective and robust approach is introduced to acquire atmospheric light value. Finally, the clear image can be restored with atmospheric scattering model. Several experimental results on synthetic and real-world images show that the proposed algorithm can protect and recover the detailed information excellently. The performance on synthetic datasets I-HAZE, O-HAZE and HazeRD are 0.762/16.598 (SSIM/PSNR),0.787/17.193, and 0.783/17.104, respectively.

Modelling and Numerical Solution of the Problem with Unilateral Constraints and Friction under Dynamic Action of the Load

Problems with unilateral constraints are not uncommon in the practice of calculating building construction and structures. Certain difficulties in solving them arise during contact friction, as well as the dynamic action of the load. It is known that such problems from a mathematical point of view s are not correct enough, their solution becomes more complicated and depends on the history of loading and deformation of the structure. At the same time, the ability to take into account the complex working conditions of the structure makes its calculation more complete and accurate. The paper considers the solution of a dynamic contact problem on the basis of the finite element method and the step-by-step loading method. Unilateral constraints with Coulomb friction are modeled using contact finite elements of a frame-rod type. The method of compensating loads is applied in order to comply with the limitations under ultimate friction-sliding conditions. Based on the considered discrete contact model and the step-by-step analysis method, a numerical algorithm has been developed, which allows in one step-by-step process to integrate simultaneously the equations of motion and implement contact conditions with Coulomb friction. With the help of the proposed approach, numerical solutions of the problem pertaining to a structure contact with the base have been obtained and analyzed at various parameters of dynamic load. Comparison of the results with the solution obtained by the well-known iteration method on the ultimate friction forces allows to conclude about the efficiency and reliability of the developed algorithm under complex contact conditions and dynamic loading.

Frictional interactions for non‐localized beam‐to‐beam and beam‐inside‐beam contact

AbstractThis contribution presents the extensions of beam‐to‐beam and beam‐inside‐beam contact schemes of the same authors towards frictional interactions. Since the schemes are based on the beams' true surfaces (instead of surfaces implicitly deduced from the beams' centroid lines), the presented enhancements are not only able to account for frictional sliding in the beams' axial directions, but also in the circumferential directions. Both the frictional beam‐to‐beam approach as well as the frictional beam‐inside‐beam approach are applicable to shear‐deformable and shear‐undeformable beams, as well as to beams with both circular and elliptical cross‐sections (although the cross‐sections must be rigid). A penalty formulation is used to treat unilateral and frictional contact constraints. FE implementation details are discussed, where automatic differentiation techniques are used to derive the implementations. Simulations involving large sliding displacements and large deformations are presented for both beam‐to‐beam and beam‐inside‐beam schemes. All simulation results are compared with those of the frictionless schemes.

A Class of Nonlinear Inclusions and Sweeping Processes in Solid Mechanics

We consider a new class of inclusions in Hilbert spaces for which we provide an existence and uniqueness result. The proof is based on arguments of monotonicity, convexity and fixed point. We use this result to establish the unique solvability of an associated class of Moreau’s sweeping processes. Next, we give two applications in Solid Mechanics. The first one concerns the study of a time-dependent constitutive law with unilateral constraints and memory term. The second one is related to a frictional contact problem for viscoelastic materials. For both problems we describe the physical setting, list the assumptions on the data and provide existence and uniqueness results.

Gauss optimization method for the dynamics of unilateral contact of rigid multibody systems

The discontinuous dynamical problem of multi-point contact and collision in multi-body system has always been a hot and difficult issue in this field. Based on the Gauss’ principle of least constraint, a unified optimization model for multibody system dynamics with multi-point contact and collision is established. The paper presents the study of the numerical solution scheme, in which particle swarm optimization method is used to deal with the corresponding optimization model. The article also presents the comparison of the Gauss optimization method (GOM) and the hybrid linear complementarity method (i.e. combining differential algebraic equations (DAEs) and linear complementarity problems (LCP)), commonly used to solve the dynamic contact problem of multibody systems with bilateral constraints. The results illustrate that, the GOM has the same advantage of dynamical modelling with LCP and when the redundant constraint exists, the GOM always has a unique solution and so no additional processing is needed, whereas the corresponding DAE-LCP method may have singular cases with multiple solutions or no solutions. Using numerical examples, the GOM is verified to effectively solve the dynamics of multibody systems with redundant unilateral and bilateral constraints without additional redundancy processing. The GOM can also be applied to the optimal control of systems in the future and combined with the parameter optimization of systems to handle dynamic problems. The work given provides the dynamics and control of the complex system with a new train of thought and method. The paper establishes a unified optimization model and the corresponding numerical solution scheme for multibody system dynamics with multi-point contact and collision based on Gauss’ principle of least constraint. Comparisons between Gauss optimization method (GOM) and the hybrid linear complementarity method (HLCM) are carried out through theoretical analysis and numerical examples, which show that GOM has the same advantage of dynamical modelling as HLCM and when redundant constraints exist, GOM has unique solutions without additional processes, whereas HLCM may have singular cases with multiple or no solutions.

A Remark on Elastic Graphs with the Symmetric Cone Obstacle

This paper is concerned with the variational problem for the elastic energy defined on symmetric graphs under the unilateral constraint. Assuming that the obstacle function satisfies the symmetric cone condition, we prove (i) uniqueness of minimizers and (ii) loss of regularity of minimizers and give (iii) complete classification of existence and nonexistence of minimizers in terms of the size of obstacle. As an application, we characterize the solution of obstacle problem as equilibrium of the corresponding dynamical problem.

Numerical Analysis of a Sliding Frictional Contact Problem with Normal Compliance and Unilateral Contact

This paper represents a continuation of [1] and [2]. Here, we consider the numerical analysis of a non-trivial frictional contact problem in a form of a system of evolution nonlinear partial differential equations. The model describes the equilibrium of a viscoelastic body in sliding contact with a moving foundation. The contact is modeled with a multivalued normal compliance condition with memory term restricted by a unilateral constraint and is associated with a sliding version of Coulomb’s law of dry friction. After a description of the model and some assumptions, we derive a variational formulation of the problem, which consists of a system coupling a variational inequality for the displacement field and a nonlinear equation for the stress field. Then, we introduce a fully discrete scheme for the numerical approximation of the sliding contact problem. Under certain solution regularity assumptions, we derive an optimal order error estimate and we provide numerical validation of this result by considering some numerical simulations in the study of a two-dimensional problem.

State-constrained controllability of linear reaction-diffusion systems

We study the controllability of a coupled system of linear parabolic equations, with nonnegativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an “approximate” nonnegativity constraint, and a another stronger one, with “exact” nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a “staircase” method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state.

Нелинейная задача о межматериальной трещинепод воздействием волны сдвига

Solution of the problem for an interface crack under the action of a harmonic shear wave is presented. It is shown that the same problems solutions of other authors were performed without taking into account the crack faces contact, and results obtained indicate the interpenetration of the faces, that is not possible. Thus, it is proved that the problem is nonlinear because the positions and sizes of the contact zone are unknown and variable during the loading. The solution is obtained by the boundary integral equations method taking into account the contact interaction of the crack faces: using the Somigliana dynamic identity and the boundary equations arising from them, the transition from the two-dimensional problem to the equivalent problem at the boundaries of the domain is realized; the vector components in the boundary integral equations are presented by Fourier series, to prevent the interpenetration of the crack faces and the emergence of tensile forces in the contact zone the Signorini unilateral constraints are involved. The numerical solution is performed by the method of boundary elements with constant approximation of the problem parameters on an element. Numerical researches of the shear wave frequency influence onto the crack faces and adjoining surface displacements, opening and extent of crack faces contact zone are carried out. The quantitative difference between the maximum tangential and normal components of adhesion line and the crack faces displacements is shown. It is shown that the position and length of the contact area change during the load period, and the magnitudes of the contact forces vary along the crack length.

Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints

This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems.

Biot-pressure system with unilateral displacement constraints

A fully saturated poroelastic medium is confined by the sides of a cylinder, and the regions below and above the medium are filled with fluid at respective constant pressures. The filtration flow of fluid through the poroelastic medium and the small deformations of the medium are described by a quasi-static Biot system of partial differential equations. Our objective here is to establish the well-posedness of an initial-boundary-value problem for this system in which the poroelastic medium is fixed and sealed on the sides, free and in contact with the exterior fluid on the top and bottom, and displacement of the medium is unilaterally constrained on the top by a Signorini-type free boundary condition.