Stability estimates in identification problems for the convection-diffusion-reaction equation

Identification problems for the stationary convection-diffusion-reaction equation in a bounded domain with a Dirichlet condition imposed on the boundary of the domain are studied. By applying an optimization method, these problems are reduced to inverse extremum problems in which the variable diffusivity and the volume density of substance sources are used as control functions. Their solvability is proved for an arbitrary weakly lower semicontinuous cost functional and particular cost functionals. An analysis of the optimality system is used to establish sufficient conditions on the input data under which the solutions of particular extremum problems are unique and stable with respect to small perturbations in the cost functional and in one of the functions involved in the boundary value problem.     

On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.     

Discrete boundary smoothing using control node parameterisation for aerodynamic shape optimisation

This paper presents an automated aerodynamic optimisation algorithm using a novel method of parameterising the search domain and geometry by employing user–defined control nodes. The displacement of the control nodes is coupled to the shape boundary movement via a ‘discrete boundary smoothing’. This is initiated by a linear deformation followed by a discrete smoothing step to act on the boundary during the mesh movement based on the change in its second derivative. Implementing the discrete boundary smoothing allows both linear and non-linear shape deformation along the same boundary dependent on the preference of the user. The domain mesh movement is coupled to the shape boundary movement via a Delaunay graph mapping. An optimisation algorithm called Modified Cuckoo Search (MCS) is used acting within the prescribed design space defined by the allowed range of control node displacement. In order to obtain the aerodynamic design fitness a finite volume compressible Navier-Stokes solver is utilized. The resulting coupled algorithm is applied to a range of case studies in two dimensional space including the optimisation of a RAE2822 aerofoil and the optimisation of an intake duct under subsonic, transonic and supersonic flow conditions. The discrete mesh–based optimisation approach outlined is shown to be effective in terms of its generalised applicability, intuitiveness and design space definition.     

Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

We deal with the problem of estimating the volume of inclusions using a small number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq–Vogelius in the limit when the volume fraction tends to zero. The bounds of this article, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.     

An efficient iteration approach for nonlinear boundary value problems in 2D piezoelectric semiconductors

Based on initial nonlinear constitutive equations, we establish the extended displacement and traction boundary integral equations for a piezoelectric medium with a volume electric charge, along with electron and electric current density boundary integral equations for a conductor with a volume electric current. Then, an iterative approach is proposed for investigation of boundary value problems in two-dimensional piezoelectric semiconductors (PSCs). Compared with extended displacements obtained by finite element analysis, this approach is validated via a rectangular PSC under extended external loads. Furthermore, as a numerical example, extended displacements across an elliptical hole in a rectangular PSC are investigated. It is shown that there is a stress concentration near the elliptical hole, which is closely dependent on its shape.     

On box schemes for elartliptic variational inequalities

General finite volume approximations in two and three space dimensions are studied for the discretization of interior and boundary obstacle problems with mixed boundary conditions. First order convergence of the finite volume (box) solution is shown in the energy norm. Based on a discrete maximum principle there are proposed two penalization techniques for the solution of the finite volume inequalities. By coupling of discretization and penalty parameters the overall error is analyzed. The iterative solution of the penalty problems is discussed. Finally, results of numerical experiments are presented to illustrate the convergence behaviour between the exact, the box and the penalty solutions.     

Static response and free vibration analysis of FGM plates using higher order shear deformation theory

Free vibration and static analysis of functionally graded material (FGM) plates are studied using higher order shear deformation theory with a special modification in the transverse displacement in conjunction with finite element models. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. The fundamental equations for FGM plates are derived using variational approach by considering traction free boundary conditions on the top and bottom faces of the plate. Results have been obtained by employing a continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present model. Numerical results for different thickness ratios, aspect ratios and volume fraction index with different boundary conditions have been presented. It is observed that the natural frequency parameter increases for plate aspect ratio, lower volume fraction index n and smaller thickness ratios. It is also observed that the effect of thickness ratio on the frequency of a plate is independent of the volume fraction index. For a given thickness ratio non-dimensional deflection increases as the volume fraction index increases. It is concluded that the gradient in the material properties plays a vital role in determining the response of the FGM plates.     

Limit boundary conditions for finite volume approximations of some physical problems

In the industrial context, finite volume schemes are used to compute an approximation of the solution of a system of equations set on a certain domain. When this domain is bounded, some numerical boundary conditions have to be implemented in order to complete the computation of the finite volume scheme. This is a tricky step in the elaboration of the scheme, which is still not mastered. In fact, at a closer sight, it appears that there is a deep interaction between the understanding of the physical phenomena at the boundary of the domain and the implementation of the numerical boundary conditions. Unfortunately, this link is not always completely intelligible and a reason for this lack of clarity is the fact that, whereas the continuous equation satisfied by the limit of the numerical solution is known, the boundary conditions satisfied by this very limit are not well-understood. The purpose of this paper is to clarify this point in three industrial situations of one-dimensional two-phase flows.     

Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling

We consider a model system consisting of two reaction–diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division.Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction–diffusion system.     

Asymptotic analysis of a boundary optimal control problem on a general branched structure

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar‐type (non‐smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch.     

Analytical solution for the evolution of exit point along the sloping seepage face

This paper developed an analytical solution for the problem of exit point evolution on the seepage face in the unconfined aquifer with sloping interface. A theoretical model for the groundwater drawdown problem in a half‐infinite aquifer with a sloping boundary is built in accordance with the linearized one‐dimensional Boussinesq equation and the Neumann boundary condition at the seepage point. The homotopy analysis method is then adopted for solving this dynamic boundary problem. By constructing two continuous deformations, the original problem could be converted into a group of subproblems with the same physical essence and similar mathematical solutions. To compare this analytical solution, a numerical model based on the finite volume method is developed, which employs adaptive grids to settle the dynamic boundary condition. The comparisons show that the analytical solution agrees with the numerical model well. The results are useful for the quantification of various hydrological problems. The methodology applied in this study is referential for other dynamic boundary problems as well.     

Two-parameter extremum problems of boundary control for stationary thermal convection equations

Two-parameter extremum problems of boundary control are formulated for the stationary thermal convection equations with Dirichlet boundary conditions for velocity and with mixed boundary conditions for temperature. The cost functional is defined as the root mean square integral deviation of the desired velocity (vorticity, or pressure) field from one given in some part of the flow region. Controls are the boundary functions involved in the Dirichlet condition for velocity on the boundary of the flow region and in the Neumann condition for temperature on part of the boundary. The uniqueness of the extremum problems is analyzed, and the stability of solutions with respect to certain perturbations in the cost functional and one of the functional parameters of the original model is estimated. Numerical results for a control problem associated with the minimization of the vorticity norm aimed at drag reduction are discussed.     

Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.     

Equilibrium failure processes of granular composites

The macroscopic failure of inhomogeneous media results from damage accumulation on different structural levels. During rigid loading, when given displacements of boundary points are ensured, irrespective of the body's resistance, structural-failure processes of composite materials take place in an equilibrium regime and result in the manifestation of such nonlinear-behavior effects as a descending branch on the strain diagram. the structural elements of a granular composite are homogeneous and firmly connected along the interface. Their geometry and mutual arrangement are given and do not change during deformation and failure of the medium, and the medium itself is macrohomogenous. The strength of isotropic structural elementsis estimated by comparing the second invariant of the stress tensor with its critical value. Nonfulfillment of the indicated strength criterion is associated with loss of ability to resist changes in form; at this point, the positive value of the first invariant corresponds to loss of such ability to resist and increase in volume. The deformation and structural failure of the medium are investigated as a single process that can be described under quasi-static loading by a boundary problem consisting of a closed system of Eqs. (1) and (2) and boundary conditions providing for a macrohomogeneous strain state. A principal feature of the boundary problem under consideration is the possibility of considering in constitutive relationships the states of the inhomogeneous medium, which correspond to partial or complete loss of bearing capacity of the structural elements. The random structural strength constants correspond to three-parameter Weibull distribution (6). The representative volume of a granular composite, which fills a domain in the form of a cube, is modeled by a set of istropic elastobrittle strain diagrams containing a descending branch are obtained as a result of the mathematical modeling of deformation processes and structural failure to realized a representative volume containing 384 structural elements with different strength and similar elastic constants.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Perm'State Mechanical University, Russia. Translated From Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 808–817, November–December, 1996.     

Optimal Shape Problems for Eigenvalues

ABSTRACT

We consider the first Dirichlet eigenvalue for nonhomogeneous membranes. For given volume we want to find the domain which minimizes this eigenvalue. The problem is formulated as a variational free boundary problem. The optimal domain is characterized as the support of the first eigenfunction. We prove enough regularity for the eigenfunction to conclude that the optimal domain has finite parameter. Finally an overdetermined boundary value problem on the regular part of the free boundary is given.     

A note on the stability of cut cells and cell merging

Embedded boundary meshes may have cut cells of arbitrarily small volume which can lead to stability problems in finite volume computations with explicit time stepping. We show that time step constraints are not as strict as often believed. We prove this in one dimension for linear advection and the first order upwind scheme. Numerical examples in two dimensions demonstrate that this carries over to more complicated situations. This analysis sheds light on the choice of time step when using cell merging to stabilize the arbitrarily small cells that arise in embedded boundary schemes.     

Stochastic shape sensitivity in powder metallurgy processing

Stochastic shape sensitivity in forming process of powder metallurgy materials is analyzed. For this purpose the rigid-poroplastic material model has been assumed. The theoretical formulation for stochastic shape sensitivity is described which presents probabilistic distributions taking into account random initial and boundary conditions. The control volume approach is discussed. Stochastic finite element equations for rigid – poroplastic materials are solved for the first two probabilistic moments. Numerical simulations were performed to illustrate shape sensitivity problems in the process of compression of rigid-poroplastic cylinder. The differences in deterministic and stochastic sensitivities are presented. The results derived can be used for the subsequent quantitative stochastic shape design as well as stochastic shape optimization.     

Partial surface matching by using directed footprints

In this paper we present a new technique for partial surface and volume matching of images in three dimensions. In this problem, we are given two objects in 3-space, each represented as a set of points, scattered uniformly along its boundary or inside its volume. The goal is to find a rigid motion of one object which makes a sufficiently large portion of its boundary lying sufficiently close to a corresponding portion of the boundary of the second object. This is an important problem in pattern recognition and in computer vision, with many industrial, medical, and chemical applications. Our algorithm is based on assigning a directed footprint to every point of the two sets, and locating all the pairs of points (one of each set) whose undirected components of the footprints are sufficiently similar. The algorithm then computes for each such pair of points all the rigid transformations that map the first point to the second, while making the respective direction components of their footprints coincide. A voting scheme is employed for computing transformations which map significantly large number of points of the first set to points of the second set. Experimental results on various examples are presented and show the accurate and robust performance of our algorithm.     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilization of an Elastic Plate with Viscoelastic Boundary Conditions

The boundary control problem of an elastic thin plate with boundary viscoelasticity is formulated in the standard form of a linear infinite-dimensional systems in the energy Hilbert space. The feedback control is designed so that the input and output are collocated. The frequency-domain approach is adopted in investigating the exponential stability of the closed-loop system. Finally, by considering the boundary viscoelasticity as damping, we establish a strong stability result based on the LaSalle invariance principle and the Hömander uniqueness theorem.