On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation.

A Variational Method for Multivalued Boundary Value Problems

In this paper, we investigate the existence of solution for differential systems involving a ϕ−Laplacian operator which incorporates as a special case the well-known p−Laplacian operator. In this purpose, we use a variational method which relies on Szulkin’s critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive.

Questions on Optimization of Izomorphic-Holonomic Information-Measuring Systems

Questions on optimization of isomorphic-holonomic information –measuring systems, characterized by in-ternal holomorphic relation are considered. It is shown that isomorphic-holonomic property of information measuring and mechatronic systems make it possible to carry out optimization of them transforming of this task to Lagrange task where the target functional (Lagrange functional) is sum of initial target functional and integral of function of holonomic relation function with limitation imposed on it multiplied by Lagrange multi-plier. It is proved that if searched for function of holonomic relation with imposed limitation condition pro-vides for minimum (maximum) of target functional and if inegrant of initial target functional can be linearized so the function is always exists and is featured as inversed to link function upon which maximum (minimum) can be achieved.

Numerical analysis of the stress-strain state of thin shells based on a joint triangular finite element

Relevance. The use of the finite element method for determining the stressstrain state of thin-walled elements of engineering structures predetermines their discretization into separate finite elements. Splitting irregular parts of the structure is impossible without the use of triangular areas. The triangular elements of shell structures are joint in displacements and in their derivatives only at the nodal points. Therefore, ways to improve the compatibility conditions at the boundaries of triangular elements are relevant. Aims of research. The aim of the work is to improve the compatibility conditions at the boundaries of adjacent triangular elements based on equating the derivatives of normal displacements in the middle of the boundary sides. Methods. In order to improve the compatibility conditions at the boundaries of triangular elements in this work, the Lagrange functional is used with the condition of ensuring equality in the middle of the sides of adjacent elements derived from normal displacements in the directions of perpendiculars tangent to the middle surface of the shell. Results. Using the example of analysing an elliptical shell, the efficiency of using a joint triangular finite element is shown, whose stiffness matrix is formed in accordance with the algorithm outlined in this article.

Mathematical Modeling of the Contact Interaction of Two Elastic Bodies Using the Mortar Method

The article discusses the development of an algorithm for solving contact problems of elasticity theory. Solving such problems is often associated with necessity of using mismatched grids. Their joining can be carried out both with the help of iterative procedures that form the so-called Schwarz alternating methods, and with the help of the Lagrange multipliers method or the penalty method. The algorithm constructed in the article uses the mortar method for matching the finite elements on the contact line. All these methods of joining the grids make it possible to ensure continuity of displacements and stresses near the contact line. However, one of the main advantages of the mortar method is the possibility of independent choice of different types of finite elements and form functions both on both boundaries of two bodies on the contact line, and when integrating along it. The application of this method in conjunction with the classical formulation of the finite element method based on the minimization of the Lagrange functional leads to a system of linear algebraic equations with a saddle point. The article discusses in detail its numerical solution based on the modified symmetric successive upper relaxation method.The results of the constructed algorithm are demonstrated on three test contact problems. They analyze the stress-strain state of differently loaded contacting two-dimensional plates. The examples considered show that continuity of the displacements of displacements and stresses is preserved near the contact line. The versatility of the developed algorithm leaves the possibility of further analysis of the effectiveness of the mortar method using different types of finite elements and form functions.

A FIXED POINT APPROACH TO THE STABILITY OF GENERAL QUADRATIC EULER-LAGRANGE FUNCTIONAL EQUATIONS IN INTUITIONISTIC FUZZY SPACES

In this paper, we prove the generalized Hyers-Ulam stability of a general k-quadratic Euler-Lagrange functional equation:for any fixed positive integer in intuitionistic fuzzy normed spaces using a fixed point method.

Superlinear elliptic problems under the non-quadraticity condition at infinity

We present some sufficient conditions to obtain compactness properties for the Euler–Lagrange functional of an elliptic equation. As an application, we extend some existence and multiplicity results for superlinear problems.

The Application of Shape Gradient for the Incompressible Fluid in Shape Optimization

This paper is concerned with the numerical simulation for shape optimization of the Stokes flow around a solid body. The shape gradient for the shape optimization problem in a viscous incompressible flow is evaluated by the velocity method. The flow is governed by the steady-state Stokes equations coupled with a thermal model. The structure of continuous shape gradient of the cost functional is derived by employing the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. A gradient-type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose, and the proposed algorithm is feasible and effective.