Mixed Variational Problems Research Articles

Stability of a three-layer cylindrical shell

Problem. The article presents the results of studies of the stability of a three-layer cylindrical shell. Factors that have a significant impact on the strength and stability of the three-layer cylindrical shell were taken into account, namely the reduced modules of elasticity of the three-layer wall. Goal. The purpose of the study is to develop a methodology for calculating the stability of a three-layer cylindrical shell, which will significantly improve the calculation practice of such structures in relation to determining the critical external pressure for a three-layer cylindrical shell. Methodology. The method of variational calculation using the Euler equation of the mixed variational problem was used to determine the critical pressure. To determine the stability of the three-layer cylindrical shell, those factors that have a significant impact on its strength and stability were taken into account, namely the reduced modulus of elasticity of the three-layer wall. The bending stiffness Dh was replaced by the bending stiffness of the three-layer shell taking into account the shear deformation. Results. The current state of the issue of the stability of a three-layer cylindrical shell has been studied. Using the methods of variational calculation with the Euler equation of the mixed variational problem, the equation of the condition of equality of the internal and external forces of the orthotropic structure, which is in a state of indifferent equilibrium with radial displacement, is compiled. Accepting the previously obtained equation for radial movements and substituting it into the equation for the potential energy of the system per unit length, we obtained the equation for determining the critical pressure. The obtained analytical solution was tested for the design of the crane running wheel, which has an elastic insert. We will get pkr=1267 MPa. For crane running wheels, the allowable pressure of the wheel on the rail is taken within 250 MPa, that is, we have a reserve of stability of the rail nc=1267/250=5.1. As you can see, the margin of stability is more than sufficient. In addition, a comparison was made between the obtained method of calculating the stability of a three-layer cylindrical shell and the method described in [20]. Originality. A new method of calculating a three-layer cylindrical shell under the action of external pressure has been developed. A quantitative assessment of the critical pressure of the traveling crane wheel, which has an elastic insert, was carried out. Practical value A technique for determining the critical pressure of a three-layer cylindrical structure under the action of external pressure has been created.

A mixed variational inequality method for solving Signorini problems

This paper presents a new iterative method for solving Signorini problems. Boundary integral equations are established by discretizing the boundary of the problem, and the system equations related only to the unknown variables on the boundary of the problem are derived. Then, the Signorini boundary conditions of the problem are expressed as equivalent variational inequalities. By considering a simultaneous system of equations and variational inequalities, the Signorini problem is reduced to a mixed variational inequality problem. Finally, a compatibility iteration algorithm for solving mixed variational inequalities is designed based on the projection-contraction algorithm. Several examples are tested to demonstrate the accuracy and effectiveness of the proposed algorithm.

Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions

In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results.

Existence and Approximation of Solution for Vector Mixed Variational-Like Inequality Under Densely (η, f)-C-Pseudomonotone Mappings

In this paper we study vector mixed variational inequality problem under (η, C)-pseudomonotonicity and densely (η, f)-C-pseudo-monotonocity in reflexive Banach space. The existence and uniqueness of solutions have been established with the help of KKM technique and further we have proposed iterative algorithm to find the approximate solution of vector mixed variational inequality by defining an auxiliary problem.

Determination of the stability of a three-layer shell of a traveling wheel with light filler

Purpose. Development of a calculation methodology for three-layer cylindrical shell stability, which will significantly improve the calculation practice for these structures regarding the determination of critical external pressure. Methodology. When determining the critical external pressure, the method of variational calculation using the Euler equation of the mixed variational problem was used. To determine three-layer cylindrical shell stability, the factors of significant influence on its strength and stability were taken into consideration, namely the reduced modulus of a three-layer wall elasticity. Bending stiffness Dh was substituted with flexural stiffness of three-layer shell with account for the shear deformation. Findings. The current situation of the three-layer cylindrical shell stability issue is investigated. Using the variational calculation methods via the Euler equation of the mixed variational problem an equation is composed of equality condition of inner and external force action of an orthotropic structure, which is under the state of neutral equilibrium with radial displacement. The previously obtained equation for radial displacements having been taken and applied to the system potential energy per unit of length equation, an equation for the critical pressure determination is determined. The analytical solution obtained was proposed for the structure of the crane travelling wheel with an elastic insert. Pcr 1267MPa was obtained. The allowable wheel pressure on rail for the crane travelling wheels is adopted to be within 250 MPa, i.e. the available stability margin is nc 1267/250 5.1. As we can see, the stability margin is more than sufficient. Originality. A new methodology for the three-layer cylindrical shell under external pressure calculation is developed. A quantitative assessment of the crane travelling wheel with flexible insert critical pressure is carried out. Practical value. A determination methodology for critical pressure of a three-layer cylindrical structure under external pressure is created.

Two fast variance-reduced proximal gradient algorithms for SMVIPs-Stochastic Mixed Variational Inequality Problems with suitable applications to stochastic network games and traffic assignment problems

In this paper, we propose two proximal gradient algorithms with variance reduction for stochastic mixed variational inequality problems. One is a proximal extragradient algorithm and another is a proximal forward–backward–forward algorithm. Under the monotonicity assumption on the mapping F and other moderate conditions, we derive some asymptotic convergence properties and O(1/k) convergence rate in terms of the restricted gap function values for the proposed algorithms. Furthermore, under the bounded metric subregularity condition, we investigate the linear convergence rate and oracle complexity bounds for the proposed algorithms when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate of ⌈k+1⌉−s with s>0, the mean-squared distance of the iterates to the solution set decays at a corresponding polynomial rate, while the iterations and oracle complexities to obtain an ε-solution are O(1/ε1/s) and O(1/ε1+1/s) respectively. Finally, some numerical experiments on stochastic network games and traffic assignment problems indicate that the proposed algorithms are efficient.

Coupled Domain-Boundary Variational Formulations for Hodge\u2013Helmholtz Operators

We couple the mixed variational problem for the generalized Hodge–Helmholtz or Hodge–Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a “rough” surface. The low-frequency robustness of the potential formulation of Maxwell’s equations makes this model a promising starting point for Galerkin discretization.

Mixed Variational Inequality Problem Involving Generalized Yosida Approximation Operator in q‐Uniformly Smooth Banach Space

A mixed variational inequality problem involving generalized Yosida approximation operator is considered and studied in q‐uniformly smooth Banach space. We have shown that mixed variational inequality problem involving generalized Yosida approximation operator is equivalent to a fixed‐point equation. The fixed‐point formulation is applied to establish an algorithm to obtain the solution of mixed variational inequality problem involving generalized Yosida approximation operator. Convergence criteria are also discussed. In support of our main result, we provide an example using Matlab program together with a computation table and convergence graphs. To check the validity of mixed variational inequality problem involving generalized Yosida approximation operator and its fixed‐point formulation, we construct one more example.

A coercive mixed formulation for the generalized Maxwell problem

A coercive mixed variational formulation on H0(curl;Ω)×H(div;Ω) is proposed for the generalized Maxwell problem which typically arises from computational electromagnetism. The mixed variables are the electric field and a pseudo electric displacement field. The well-posedness of the mixed variational problem is proven in the general settings (multiply connected domain of Lipschitz-continuous boundary with a number of connected components, filling with discontinuous, anisotropic and inhomogeneous media); more importantly, the coercivity is established. A conforming finite element discretization is further proposed, where the electric field is approximated by H(curl;Ω)-conforming edge element while the pseudo electric displacement field by H(div;Ω)-conforming flux element. Error estimates are obtained, and in particular, the method produces an L2 curl-convergent approximation and more importantly, an L2 div-convergent approximation for the solution.

Stability and Convergence Analysis for Set-Valued Extended Generalized Nonlinear Mixed Variational Inequality Problems and Generalized Resolvent Dynamical Systems

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.

Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings

Abstract In this paper, we introduce a shrinking projection method of an inertial type with self-adaptive step size for finding a common element of the set of solutions of a split generalized equilibrium problem and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step size incorporated helps to overcome the difficulty of having to compute the operator norm, while the inertial term accelerates the rate of convergence of the proposed algorithm. Under standard and mild conditions, we prove a strong convergence theorem for the problems under consideration and obtain some consequent results. Finally, we apply our result to solve split mixed variational inequality and split minimization problems, and we present numerical examples to illustrate the efficiency of our algorithm in comparison with other existing algorithms. Our results complement and generalize several other results in this direction in the current literature.

Error Bounds for Inverse Mixed Quasi-Variational Inequality via Generalized Residual Gap Functions

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.

Strong and weak convergence theorems for general mixed equilibrium problems and general variational inequality problems and fixed point problems for two nonexpansive semigroups in Hilbert spaces

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of the general mixed equilibrium problem and the set of solutions of a general variational inequality for two cocoercive mappings and the set of common fixed points of two nonexpansive semigroups in Hilbert space. We obtain both strong and weak convergence theorems for the sequences generated by these iterative processes in Hilbert spaces. Our results improve and extend the results announced by many others.

ON A CLASS OF SADDLE POINT PROBLEMS AND CONVERGENCE RESULTS

We consider an abstract mixed variational problem consisting of two inequalities. The first one is governed by a functional φ, possibly non-differentiable. The second inequality is governed by a nonlinear term depending on a non negative parameter ǫ. We study the existence and the uniqueness of the solution by means of the saddle point theory. In addition to existence and uniqueness results, we deliver convergence results for ǫ → 0. Finally, we illustrate the abstract results by means of two examples arising from contact mechanics.

On the Weak Solvability Via Lagrange Multipliers for a Bingham Model

We study the stationary flow of an incompressible non-Newtonian fluid of Bingham type, mathematically described by means of a nonlinear boundary value problem governed by PDEs. The variational formulation which we propose is a mixed variational problem with Lagrange multipliers. First, we obtain existence, uniqueness, and stability results into an abstract framework. Then, we discuss the well-posedness of the mechanical model based on the auxiliary abstract results.

Tykhonov well-posedness of a mixed variational problem

ABSTRACT We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem have already been obtained in the literature. In this current paper we complete these results by proving the well-posedness of the problem, in the sense of Tykhonov. To this end we introduce a family of approximating problems for which we state and prove various equivalence and convergence results. We illustrate these abstract results in the study of a frictionless contact model with elastic materials. The process is assumed to be static and the contact is with unilateral constraints. We derive a weak formulation of the model which is in the form of a mixed variational problem with unknowns being the displacement field and the Lagrange multiplier. Then, we prove various results on the corresponding mixed problem, including its well-posedness in the sense of Tykhonov, under various assumptions on the data. Finally, we provide mechanical interpretation of our results.

Isotonicity of the proximity operator and mixed variational inequalities in Hilbert spaces

In this paper, the isotonicity of the proximity operator and its applications are discussed. We first establish a few new conditions of the mappings such that their proximity operators are isotone with respect to orders induced different minihedral cones. Some properties and examples for these conditions are then introduced. We especially consider the isotonicity of the proximity operator with respect to one order induced by a subdual cone and two orders. To estimate the convergence rate of the iterative algorithms, some other inequality characterizations of the proximity operator with respect to the orders are then proved. As applications, some solvability and approximation theorems for the mixed variational inequality and optimization problems are established by order approaches, in which the mappings need not to be continuous and the solutions are optimal with respect to the orders. By using the isotonicity of the proximity operator with respect to two orders, we overcome the absence of the regularity of the order. The convergence rate of forward–backward algorithms is finally estimated by order approaches.

Notes on a saddle point reformulation of mixed variational problems

We summarize some general ideas regarding approximation of mixed variational problems using saddle point reformulation. We consider the concepts of optimal and almost optimal (or α) test norm and provide estimates for the continuity and stability constants. A preconditioning strategy for solving the discrete mixed formulations is used in combination with the special test norms. We further provide a choice for a discrete trial space, that depends on the choice of a standard test space and leads to discrete stability, when using the appropriate test norm. Examples to illustrate how the stability of the saddle point discretization can be improved using special test norms are included.

On the Adaptive Proximal Method for a Class of Variational Inequalities and Related Problems

For problems of unconstrained optimization, the concept of inexact oracle proposed by O. Devolder, F. Glineur and Yu.E. Nesterov is well known. We introduce an analog of the concept of inexact oracle (model of a function) for abstract equilibrium problems, variational inequalities, and saddle-point problems. This allows us to propose an analog of Nemirovskii’s known mirror prox method for variational inequalities with an adaptive adjustment to the smoothness level for a fairly wide class of problems. The auxiliary problems at the iterations of the method can be solved with error. It is shown that the resulting errors do not accumulate during the operation of the method. Estimates of the convergence rate of the method are obtained, and its optimality from the viewpoint of the theory of lower oracle estimates is established. It is shown that the method is applicable to mixed variational inequalities and composite saddle-point problems. An example showing the possibility of an essential acceleration of the method as compared to the theoretical estimates due to the adaptivity of the stopping rule is given.

Using the generalized collage theorem for estimating unknown parameters in perturbed mixed variational equations

In this paper, we study a mixed variational problem subject to perturbations, where the noise term is modelled by means of a bilinear form that has to be understood to be “small” in some sense. Indeed, we consider a family of such problems and provide a result that guarantees existence and uniqueness of the solution. Moreover, a stability condition for the solutions yields a Generalized Collage Theorem, which extends previous results by the same authors. We introduce the corresponding Galerkin method and study its convergence. We also analyze the associated inverse problem and we show how to solve it by means of the mentioned Generalized Collage Theorem and the use of adequate Schauder bases. Numerical examples show how the method works in a practical context.