Introduction to Shape optimization Noureddine Igbida1 1Institut de recherche XLIM, UMR-CNRS 6172, Facult´edes Sciences et Techniques, Universit´ede Limoges 123, Avenue Albert Thomas 87060 Limoges, France. Email : [email protected] Preliminaries on PDE 1 Contents 1. PDE ........................................ 2 2. Some differentiation and differential operators . 3 2.1 Gradient . 3 2.2 Divergence . 4 2.3 Curl . 5 2 2.4 Remarks . 6 2.5 Laplacian . 7 2.6 Coordinate expressions of the Laplacian . 10 3. Boundary value problem . 12 4. Notion of solution . 16 1. PDE A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written model- ing). Mathematical models are used in many area, as in the natural sciences (such as physics, biology, earth science, meteorology), engineering disciplines (e.g. computer science, artificial in- telligence), in the social sciences (such as economics, psychology, sociology and political science); Introduction to Shape optimization N. Igbida physicists, engineers, statisticians, operations research analysts and economists. Among this mathematical language we have PDE. These are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. In this section, we give some basic example of elliptic partial differential equation (PDE) of second order : standard Laplacian and Laplacian with variable coefficients. These equations occur in many fields to model the conduction or stationary dissemination (time independent) of the heat, electric current, a chemical concentration, etc.. 2. Some differentiation and differential operators A differential operator is an operator defined as a function of the differentiation operators : 2.1 Gradient r is the gradient of a scalar field. It is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change. A generalization of the gradient for functions on a Euclidean space that have values in another Euclidean space is the Jacobian. A further generalization for a function from one Banach space 3 to another is the Frchet derivative. For instance, consider a room in which the temperature is Introduction to Shape optimization N. Igbida given by a scalar field, T, so at each point (x,y,z) the temperature is T(x,y,z). (We will assume that the temperature does not change over time.) At each point in the room, the gradient of T at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at a point (x,y) is H(x,y). The gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field (any conservative vector field can be expressed as the gradient of a scalar field) can be evaluated by evaluating the original scalar field at the endpoints of the curve: Z φ (q) − φ (p) = rφ · dr: L It is a generalization of the fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. 2.2 Divergence ∇· the divergence operator (denoted also by div). It measures the magnitude of a vector field’s source or sink at a given point, in terms of a signed scalar. If F (x; y; z) = (F1(x; y; z);F2(x; y; z);F3(x; y; z)); 4 then Introduction to Shape optimization N. Igbida @F @F @F ∇·F = 1 + 2 + 3 : @x @y @z More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink. Formally, ZZ F · n divF(p) = lim dS V !fpg S(V ) jV j where jV j is the volume of V; S(V ) is the boundary of V; and the integral is a surface integral with n being the outward unit normal to that surface. • Likewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss's theorem) Z I r · F d = F · dΣ Vol Vol @Vol 2.3 Curl ∇× the curl operator (or rotor and denoted also by rot or curl). It is is a vector oper- ator that describes the infinitesimal rotation of a 3−dimensional vector field. If F (x; y; z) = 5 Introduction to Shape optimization N. Igbida (F1(x; y; z);F2(x; y; z);F3(x; y; z)); then i j k @ @ @ ∇×F = @x @y @z F1 F2 F3 0 1 @F3 @F2 @F3 @F1 @F2 @F1 = @ − ; − + ; − A : @y @z @x @z @x @y The direction of the curl is the axis of rotation, as determined by the so called right-hand rule, and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. • The classical Kelvin-Stokes theorem: Z I r × F · dΣ = F · dr; Σ @Σ which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary. 2.4 Remarks Remind that in the vector field theory, we have 6 • Any given vector field A can be expressed as Introduction to Shape optimization N. Igbida A = r Φ + r × Ψ; i.e. A can always be expressed as the sum of the gradient of a scalar potential Φ and the curl of a vector potential Ψ: • r × rΦ = 0; for any given filed Φ: • r · r × Ψ = 0 for any given scalar filed Ψ: • r × r × A = r(r · A) − r2A: • r · (rΨ1 × rΨ2) = 0: 2.5 Laplacian In mathematics the Laplace operator or Laplacian is a differential operator given by the diver- gence of the gradient of a function on Euclidean space. It is usually denoted by the symbols r · r; r2 or ∆: In a Cartesian coordinate system, the Laplacian is given by N 2 X @ u ∆u(x) = 2 : i=1 @xi It is named after the French mathematician Pierre-Simon de Laplace (1749-1827), who first applied the operator to the study of celestial mechanics, where the operator gives a constant multiple of the mass distribution associated to a given gravitational potential. 7 • The Laplacian ∆u(x) of a function u at a point x; up to a constant depending on the Introduction to Shape optimization N. Igbida dimension, is the rate at which the average value of u over spheres centered at x; deviates from u(x) as the radius of the sphere grows. • The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves towards or away from some point is proportional to the Laplacian of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation. It occurs in the differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. Diffusion In the physical theory of diffusion, the Laplace operator (via Laplace's equation) arises naturally in the mathematical description of equilibrium. Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary of any smooth region Ω is 0; provided there is no source or sink within Ω : Z ru · n dS = 0; @Ω where ν is the outward unit normal to the boundary of Ω: By Green formula, Z Z divru dx = ru · ν dS = 0: Ω @Ω Since this holds for all smooth regions Ω; it can be shown that this implies divru = ∆u = 0: 8 The left-hand side of this equation is the Laplace operator. The Laplace operator itself has Introduction to Shape optimization N. Igbida a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation.