Unilateral Problem for a Viscoelastic Beam Equation Type p-Laplacian with Strong Damping and Logarithmic Source

In this manuscript, we investigate the unilateral problem for a viscoelastic beam equation of p-Laplacian type. The competition of the strong damping versus the logarithmic source term is considered. We use the potential well theory. Taking into account the initial data is in the stability set created by the Nehari surface, we prove the existence and uniqueness of global solutions by using the penalization method and Faedo-Galerkin’s approximation.

Asymptotic analysis of initial flow around an impulsively started circular cylinder using a Brinkman penalization method

The initial flow past an impulsively started translating circular cylinder is asymptotically analysed using a Brinkman penalization method on the Navier–Stokes equations. The asymptotic solution obtained shows that the tangential and normal slip velocities on the cylinder surface are of the order of $1/\sqrt {\lambda }$ and $1/\lambda$ , respectively, within the second approximation of the present asymptotic analysis, where $\lambda$ is the penalization parameter. This result agrees with the estimation of Carbou & Fabrie (Adv. Diff. Equ., vol. 8, 2003, pp. 1453–1480). Based on the asymptotic solution, the influence of the penalization parameter $\lambda$ is discussed on the drag coefficient that is calculated using the adopted three formulae. It can then be found that the drag coefficient calculated from the integration of the penalization term exhibits a half-value of the results of Bar-Lev & Yang (J. Fluid Mech., vol. 72, 1975, pp. 625–647) as $\lambda \to \infty$ .

Evaluation of Catchments’ Similarity by Penalization in the Context of Engineering Tasks—A Case Study of Four Slovakian Catchments

The present study deals with the similarity of catchments, which is a preliminary investigation before performing various water resource analyses and computations regarding other catchments, e.g., catchments’ similarity may be utilized in the context of analogous calculations of river flows in catchments without measured flows. In this paper, the penalization method of evaluating similarity is proposed; this method is appropriate for tasks in which fewer catchments are analyzed for engineering purposes. In addition to the various physical characteristics of the catchment, the “catchment’s calibrability” property is also formulated and evaluated. A methodology that used specific flows from catchments in a case study from Slovakia successfully verified the proposed penalization method. This verification confirmed that physical similarity, as evaluated using the proposed penalization methodology, also helps to identify hydrological similarity, i.e., finding the most similar catchment to a given catchment in terms of the rainfall-runoff process. Such a finding can be helpful, e.g., in the computation of the mentioned flows in ungauged catchments. Determining unmeasured flows can help to solve many engineering tasks, such as various technical calculations during the design of small reservoirs, defining the potential of a given stream for supplying irrigation, flood protection, etc.

Concentrating standing waves for Davey–Stewartson systems

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

The Coupled Volume of Fluid and Brinkman Penalization Methods for Simulation of Incompressible Multiphase Flows

In this work, we contribute to the development of numerical algorithms for the direct simulation of three-dimensional incompressible multiphase flows in the presence of multiple fluids and solids. The volume of fluid method is used for interface tracking, and the Brinkman penalization method is used to treat solids; the latter is assumed to be perfectly superhydrophobic or perfectly superhydrophilic, to have an arbitrary shape, and to move with a prescribed velocity. The proposed algorithm is implemented in the open-source software Basilisk and is validated on a number of test cases, such as the Stokes flow between a periodic array of cylinders, vortex decay problem, and multiphase flow around moving solids.

Modification of the Stamp Topological Optimization Taking into Account Cyclic Fatigue based on the Finite Element Approach

The study is devoted to optimizing the volume of stamping tools used in pressure processing processes. The relevance of the research is due to the active development of additive technologies and the possibility of producing stamping tools from plastic of optimal shape, which has an important practical significance in the manufacture of thin-walled products in the aviation and automotive industries. The purpose of the study was to carry out a mathematical formulation of the problem of topological optimization of a forming die made of a polymer material with restrictions on fatigue durability and minimum volume. The task of topological optimization was to maximize the stiffness of the die under multicyclic loading. The vector description of topological optimization was based on the finite element approach. The optimization model was built on the basis of the solid isotropic material penalization method with the introduction of additional restrictions in the model of searching for pseudo-densities of the material, taking into account the duration of the force action on the stamp under multicyclic loading. In view of the nonlinearity of the resulting system of equations, the solution of the conditional optimization problem is proposed to be carried out by constructing the Lagrange objective function and using the Lagrange multiplier method. The result of the study is the proposed approach to the topological optimization of the stamp, taking into account the multicyclic loading and restrictions on the desired volume.

Reflected stochastic partial differential equations with jumps

In this paper, we establish the existence and uniqueness of solutions of reflected stochastic partial differential equations (SPDEs) driven both by Brownian motion and by Poisson random measure in a convex domain. Penalization method plays a crucial role.