Introduction to Shape Optimization

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 modeling). 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- Introduction to Shape optimization N. Igbida 3 is the gradient of a scalar field. It is a vector field that points in the direction of the Among this mathematical language we have PDE. These are a type of differential equation, In this section, we give some basic example of elliptic partial differential equation (PDE) of telligence), in the social sciences (suchphysicists, as engineers, economics, statisticians, psychology, sociology operations and research political analysts science); and economists. i.e., a relation involving an unknowntheir function partial (or derivatives functions) with of respect several to independentformulate, those variables and and variables. thus Partial aid differential equations theas are solution the used to of, propagation problems of involvingSeemingly sound distinct functions or physical of phenomena heat, may several have electrostatics, variables;be identical such mathematical electrodynamics, governed formulations, by fluid and the thus flow, same and underlying elasticity. dynamic. second order :occur standard in Laplacian many and fields Laplacianof to with the model heat, variable the electric coefficients. conduction current, a or These chemical stationary equations concentration, dissemination etc.. (time2. independent) Some differentiation and differentialA operators differential operator is an operator defined as2.1 a function of the Gradient differentiation operators : ∇ greatest rate of increase ofA the generalization scalar of field, the and gradientEuclidean whose for space magnitude functions is is on the the a Jacobian. greatest Euclidean rate A space further of that generalization change. have for values a in function another from one Banach space Introduction to Shape optimization N. Igbida 4 , )) x, y, z . ( 3 r d · ,F ) φ ∇ L ). It measures the magnitude of a vector Z x, y, z ( 2 div ) = ,F p ) ( φ − x, y, z ) ( q 1 ( F φ ) = ( x, y, z ( F the divergence operator (denoted also by It is a generalizationspace of (generally n-dimensional) the rather fundamental than theorem just of the real calculus line. to any curve in a plane or The gradient theorem, also known assays the that fundamental theorem a of calculusexpressed line for as line the integral integrals, gradient through of afield a scalar at field) gradient the can endpoints be field of evaluated (any by the evaluating curve: conservative the original vector scalar field can be • field’s source or sink at a given point, in terms of a signed scalar. If 2.2 Divergence ∇· to another is thegiven Frchet derivative. by a For scalar instance, field, considerthat T, a the so room temperature at does in each not whichat point change the that (x,y,z) over the temperature point time.) temperature will is At show isgradient each the T(x,y,z). will point direction (We determine in the will how the temperature assume fast room,Consider rises the the a most temperature gradient quickly. surface of rises The whose T in magnitudeat height that of a above direction. the point sea is level asteepness at vector of a pointing the in point slope the (x,y) at direction is that of H(x,y). point the is steepest The given slope gradient by or of the grade H magnitude at of that the point. gradient The vector. Introduction to Shape optimization N. Igbida 5 ) = x, y, z ( F ). It is is a vector oper- curl Σ or d dS . · 3 and the integral is a surface integral n | rot F ∂z · ∂F V V, | F Vol dimensional vector field. If + ) ∂ I 2 − V ( S = ∂y ∂F ZZ } Vol + p d 1 →{ F V ∂x ∂F = ∇ · ) = lim p F ( Vol Z ) is the boundary of F ∇· V ( div V,S is the volume of | being the outward unit normal to that surface. V the curl operator (or rotor and denoted also by | n Likewise the Ostrogradsky-Gauss theorem (also known astheorem) the Divergence theorem or Gauss’s More technically, the divergence represents the volume density of the outward flux of a vector • then field from an infinitesimal volume aroundor a cooled. given point. The relevant For vector example,If field consider air for air as this is it example heated is isoutward heated the in from velocity that a of region. region thehave moving it a Therefore air will positive at the a value, expand divergence as point. negative in of the and the all region the velocity is directions region a field such is source. in called that that If a the the sink. region velocity air would Formally, field cools and points contracts, the divergence is where 2.3 Curl ∇× ator that describes the infinitesimal rotation of a 3 with Introduction to Shape optimization N. Igbida 6 .   1 ∂y ∂F − 2 , r ∂x d ∂F · , 1 F ∂z Σ ∂F ∂ I + = 3 ∂x Σ ∂F d

· − , 3 2 F ∂ F ∂z ∂z ∂F 2 ∂ ∇ × − F ∂y then Σ 3 Z , 1 i j k ∂ ∂y )) F ∂F ∂x

  = = x, y, z ( 3 F ,F ) ∇× x, y, z ( 2 ,F ) x, y, z which relates the surface integralthree-space of to the the curl line of integral a of vector the field vector over field a over surface its Σ boundary. in Euclidean The classical Kelvin-Stokes theorem: ( 1 • F ( 2.4 Remarks Remind that in the vector field theory, we have The direction of the curland is the the magnitude axis of offlow rotation, the velocity as of curl determined a is moving by the fluid,whose the then curl magnitude so the is called of curl zero right-hand is is rotation. rule, called thecorresponding irrotational. circulation If form density The the of of curl the vector is the fluid. a fieldthe form fundamental surface A represents of integral vector theorem the differentiation field of for of the vectorboundary calculus fields. curl curve. of is The a Stokes’ vector theorem, field to which the relates line integral of the vector field around the Introduction to Shape optimization N. Igbida 7 , Ψ . 2 i u 2 ∇ × ∂x ∂ N =1 X i Φ + . ∇ ) = x = ( u A . ∆ A. 2 . − ∇ can be expressed as ) . A A ∇ · ) = 0 ( 2 ∇ Ψ for any given filed Φ , In a Cartesian coordinate system, the Laplacian is given by = . A × ∇ Ψ = 0 for any given scalar filed Ψ 1 Φ = 0 or ∆ Ψ can always be expressed as the sum of the gradient of a scalar potential Φ and the 2 ∇ A ( ∇ , i.e. curl of a vector potential Ψ Any given vector field • ∇ · • ∇ · ∇ × • ∇ × ∇ × • ∇ × ∇ • 2.5 Laplacian In mathematics the Laplace operatorgence or of Laplacian the is gradient a∇ of differential · ∇ operator a given function by on the Euclidean diver- space. It is usually denoted by the symbols It is named afterapplied the the operator French mathematician tomultiple Pierre-Simon the of de study the mass of Laplace distribution celestial (1749-1827), associated mechanics, who to where a first the given gravitational operator potential. gives a constant Introduction to Shape optimization N. Igbida 8 deviates x, through the boundary u is the density at equilibrium of . u = 0 By Green formula, over spheres centered at up to a constant depending on the . u , . x, ν dS · = 0 = 0 u u ∇ dS Ω n ∂ Z = ∆ · at a point u u = it can be shown that this implies u ∇ ∇ , Ω ∂ div u dx Z ∇ div Ω Z provided there is no source or sink within Ω : , ) of a function x ( u ) as the radius of the sphere grows. x ( u is the outward unit normal to the boundary of Ω ν The Laplacian represents the flux density ofnet the rate gradient flow at of a which function.is a For chemical instance, proportional the dissolved to insymbolically, the a the fluid Laplacian resulting moves equation of towards is the or the away chemical diffusion from equation. concentration some at point that point; expressed from The Laplacian ∆ dimension, is the rate at which the average value of It occurs in the differential equations that describe many physical phenomena, such as electric • • and gravitational potentials, theand diffusion quantum equation mechanics. for heat and fluid flow,Diffusion wave propagation, In the physical theory of diffusion,in the the Laplace mathematical operator description (via of Laplace’ssome equation) equilibrium. arises quantity Specifically, naturally such if as a chemical concentration, then the net flux of where of any smooth region Ω is 0 Since this holds for all smooth regions Ω Introduction to Shape optimization N. Igbida 9 then the charge q, q dx, Ω Z = q dx, ν . Ω · Z ϕ. : ϕ = 0 = ϕ ∇ ϕ = ∆ Ω ∂ ∆ q Z ϕ dx = ∆ Ω ν it follows that Z · , is equal to the charge enclosed (in appropriate units): E Ω E ∂ Z denotes the electrostatic potential associated to a charge distribution The same approach implies that the Laplacian of the gravitational potential is the mass The left-hand side of this equation is the Laplace operator. The Laplace operator itself has ϕ where the first equality uses thepotential. fact that The the Green electrostatic formula field is now the gives gradient of the electrostatic and since this holds for all regions Ω distribution. Often the chargeunknown. (or mass) Finding the distribution potential areto function given, solving and subject Poisson’s the to equation. associated suitable potential boundary is conditions is equivalent distribution itself is given by the Laplacian of This is a consequencethe of flux Gauss’s of law. the electrostatic Indeed, field if Ω is any smooth region, then by Gauss’s law a physical interpretation for non-equilibriuma diffusion source as or the sink extent to of which chemical a concentration, point inDensity represents a associated sense to made a potential precise by theIf diffusion equation. Introduction to Shape optimization N. Igbida 10 , = 0 u = 0 by the u , then ∆ u x, d u plane. ∆ v − Ω Z xy is a function that vanishes on the − IR = x. x 2 d → f 2 d 2 | ∂y ∂ v u :Ω v + is stationary around |∇ · ∇ 2 f = 0 Ω u E Z 2 u ∂x ∂ 2 1 ∇ ∆ Ω = Z ) = f u ∆ ( ) = E εv + Conversely, if is a function, and u ( u. IR E =0 → ε | d dε :Ω u Then are the standard Cartesian coordinates of the . y and is stationary around x E fundamental lemma of calculus of variations. 2.6 Coordinate expressionsTwo of dimensions the Laplacian The Laplace operator in two dimensions is given by then boundary of Ω where the last equality follows using Green’s first identity. This calculation shows that if ∆ Energy minimization Another motivation for the Laplacian appearing in physics is that solutions to To see this, suppose in a region Ω are functions that make the Dirichlet energy functional stationary : where Introduction to Shape optimization N. Igbida 11 . 2 f 2 ∂θ ∂ ϕ 2 1 sin , 2 r . m 2 ∂ f + 2 ∂ξ . ∂z ∂   m 2 . f ξ 2 2 + 2 f ∂f ∂ϕ ∂θ ∂ 2 2 ∇ f 2 ϕ ∂z ∂ 2 1 r ∂θ + ∂ + sin 2 + n 1 2   ρ f   2 2 ∂ξ + ∂y ∂ ∂ ∂ m ∂ϕ ∂r ∂f   the zenith angle). + ): r ϕ ∂ξ 3   2 n ∂ρ ∂f f ϕ 1 2 , ξ ξ sin ρ ∂ 2 ∂x ∂r ∂ 2   r 1 r , ξ · ∇ 1 = ∂ ∂ρ ξ + m = f 1 ρ ξ   f ∆ ∇ = ∆ ∂r ∂f = 2 f r 2 ∆   ∇ ∂ ∂r 2 1 r = f ∆ represents the azimuthal angle and θ In Cartesian coordinates, In cylindrical coordinates, In spherical coordinates: In general curvilinear coordinates ( In polar coordinates, Three dimensions In three dimensions, it is commonsystems. to work with the Laplacian in a variety of different coordinate (here where summation over the repeated indices is implied. Introduction to Shape optimization N. Igbida 12 r with can be , N N IR IR ∈ ⊂ 1 rθ , − 1 N − so that it is constant = S N } 0 x S \{ N f sphere, known as the spherical 1 − − N S ?1) ∆ 2 N . 1 r ) + ∂r ∂f 1 − ∂r ∂f N 1 r ( − r ∂ ∂r N 1 an element of the unit sphere − . + 1 N θ 2 r f 2 ∂r ∂ = dimensions, with the parametrization f ∆ N is the Laplace?Beltrami operator on the ( 1 − N S dimensions As a consequence, the spherical Laplacian of a function defined on To be useful in applications, a boundary value problem should be well posed. This means − Laplacian. The two radial terms can be equivalently rewritten as N representing a positive real radius and where ∆ computed as the ordinary Laplacianalong of rays, the i.e., function homogeneous extended to of IR degree 0 3. Boundary value problem A boundary value problem iscalled a the differential equation boundary together conditions. withdifferential a equation A set which of solution also additional to satisfies restraints, the a boundary boundary conditions. value problemthat given is the a input solution toon the to the problem the input. there exists Much aproving theoretical unique that work solution, boundary which in value depends thein problems continuously field fact arising of well-posed. from partial scientific differential and equations engineering is applications devoted are to In spherical coordinates in Introduction to Shape optimization N. Igbida 13 the M ∈ ) x if we do not ( . 2 u ∗ + IR IR ⊂ ∈ τ with . (Ω) dx, u . ! 2 = 1 | . u − Ω M 2 ∂ . |∇ | 1 2 Ω u in Ω So, ∂ the membrane deforms and takes a position in Ω . |∇ , 2 Ω f ∈ = 1 + f 1 + = = 0 on ) 2 r | 2 =

u ν u · τ = 0 on u ∆ , x 1 Ω |∇ u ∆ − u normal IR Z x − ∇      f, 1 + = (      r x (Ω)) = aire − ) are small, we have M | ( 2 Assuming that the membrane has small deformation, we note x u | aire M. ( τ and | 1 x u | The tension work is proportional to the variation of of the surface of the membrane : If the boundary gives a value to the problem then it is a Dirichlet boundary condition. For If the boundary gives a value to the normal derivative of the problem then it is a Neumann If the boundary has the form of a curve or surface that gives a value to the normal derivative equilibrium position of the point example, for Laplace operator : of equilibrium boundary condition. For example apply any force. If we apply a force Since and the problem itself then it is aExample Cauchy : boundary elastic condition. membrane We consider an elastic membrane that can be identifies to a flat a region Ω Introduction to Shape optimization N. Igbida 14 and (0.1) , = 0 Ω ) is minimal, /∂ u this gives the ˜ u ( , , E Ω Ω ∂ ∈ compatible with the ∈ x x M, . ) for dx. x = 0 ( ) b ˜ u dx. f ˜ u dσ   . ) = . + x fu ( Ω ∂u ∂n ˜ in Ω u u ∂ τ + Ω 2 · ∇ | ∂ f Z u u = − ∇ = 0 in Ω |∇ τ u τ f − 1 2 = 0 on ∆ (   + u dx − u Ω Ω Z u ) ˜ Z              f ∆ is an admissible deformation of τ + ) = u u u (0) = 0 ( ∆ E : τ ( where ˜ ˜ u Ω are regular, we deduce that 0 = Φ , Z is given by the displacement of each point ) ˜ u Since the membrane is at the equilibrium, the energy ˜ u f u. + and u ( E u, f = (0.1) imply the Dirichlet problem If the boundary of the membrane is fixed, then Taking Φ • expression of the potential energy of the membrane at the equilibrium Using Green formula, we deduce that The work of the force conditions imposed on so that Assuming that Now, the constraints onof the (0.1) boundary , of the the constraints membrane on gives, through the boundary integral Introduction to Shape optimization N. Igbida 15 then we obtain the Ω of the membrane ∂ , N ⊂ Γ \ N Ω ∂ = D Ω of the membrane, then boundary . . ∂ . D Ω N Ω ˜ u dσ, ∂ ∂ in Ω   in Ω in Ω 1 non null on the boundary and (0.1) imply on on Γ f ˜ u f + f f ) on Γ 1 1 x = f f = = ( ∂u ∂n b = 0 on u = = u u τ ∆ = ∆ ∆ − ∂u ∂n on a part of the boundary Γ − on the boundary   ∂u ∂n ∂u ∂n − − u , , Ω                  2 2                                                 ∂ Z normal to IR normal to IR , , 1 1 f f is a solution to homogeneous Neumann problem u and we obtain nonhomogeneous Neumann problem and we fix themixed membrane problem on the remaining boundary Γ If the boundary of the membrane is not fixed, If we apply a force that integral of (0.1) becomes If we apply a force • • • Introduction to Shape optimization N. Igbida 16 (0.2) then there is no , Ω , ∈ Ω ∂ x (Ω) satisfying is discontinuous in Ω ∈ 2 f x ∈ C for any u f = u Starts with the PDE and rewrites it in such a way that = 0 for any ∆ − u      The most optimistic case is when there exists a function for which Somewhat surprisingly, a differential equation may have solutions which (the solution) needs to verify the PDE. u such that (0.2) is satisfied. In this kind of situation, the most standard notion , (Ω) 2 all the derivatives of the solution show up. As an illustration of the concept, consider ∈ C – Based on distributions : Classical solution : all the derivatives appearing in the”classical equation solution”. exist For and instance, satisfy a the function PDE. This is the so called is called a classical solution for the problem. Weak solution : are not differentiable upfind to such the solutions. order Indeed,may of in not the several all situations PDE; exist the andin (for derivatives a appearing some a weak in precisely solution) formulation the defined but allowsu equation which sense. one is For to nonetheless instance, deemedof if to solution satisfy for the PDE equation isweak the solution, notion appropriate of for weak differentbased solution. classes on of the There equations. notion are of many One distributions. different of definitions the of most important is • • 4. Notion of solution The notion of solution forwhich a a function PDE is a precise mathematical description that show the sense in Introduction to Shape optimization N. Igbida 17 , (0.3) (0.4) (0.5) is differentiable on Ω u null on the boundary. and integrate we obtain ξ , Ω ∂ (Ω) function (which has a null trace 1 multiply this equation by a smooth , f u. f u. L , Ω Ω in Ω Ω Z null on ∂ Z ξ = = f ξ ξ ) is a function of two real variables. Assuming = x ∆ ( · ∇ u u u u = 0 on ∆ Ω = Z ∇ − u u Ω − Z The idea is to starts with the PDE and rewrite it in such and integrate we obtain , Ω ∂ is continuously differentiable on Ω null on u ξ Ω) such that (0.4) is satisfied for any regular function However, even if thisit formulation is is sometimes inadequate morethe specifically likely existence for to and the foster lowena uniqueness enough, existence determining (high can of the regularity not problem a disfavor derivative to so solution, in that take the the into new uniqueness account formulationuniqueness, falls). the kills so a physical the In good existence phenom- general, compromise and between keeping the remove too two too is much the best kills suited the to the problem. ∂ ∗ the Laplace equation a way that only a part ofComing the derivatives back of to the the solution Laplacian of equation, the we equation assume show that up. that where Ω is a bounded domain and The new form issolutions. called Here, the a weak weak formulation, solutionon may and be the just solutions an to it are called weak multiply this equation by a smooth function that that function – Based on compromise : Introduction to Shape optimization N. Igbida 18 ξ to be u and such ξ, to be differentiable. Here, a weak solution u (Ω)) in order to use adequate tools to solve the 1 0 H ∈ u (Ω) function such that (0.5) is satisfied for any regular function 1 , 1 0 W (for instance u 1. We have shown that equation (0.3) implies equation (0.4) and/or (0.5) problem. Even more, in somesolution situations we can be brought to ask more regularity for the ∗ The new form isweak called solutions. again the weak formulation, andThe the difference solutions between to the itdifferentiable two are as formulation called to is the that secondmay in one be the we just first ask an wenull do on not the boundary. ask may not be twice differentiable and thus, they do not satisfy equation (0.3) in a u ”strong” sense. in modeling realthen world the phenomena only do waysituations not of where an admit solving equation such sufficientlyto does equations have first smooth is differentiable prove solutions solutions, using the and are it existence the in of is weak fact often weak formulation. smooth solutions convenient enough. and Even only in later show that those solutions as long as uexist functions is u differentiable. which The satisfy key either to the equation the (0.4) concept or of (0.5) weak for solution any is that there 2. Weak solutions are important because a great many differential equations encountered 3. In the case of hyperbolic systems, the notion of weak solution based on distributions Remark 0.1 Introduction to Shape optimization N. Igbida 19 does not guaranteeconditions uniqueness, or and some other it selection is criterion. necessary to supplementdefinition of it weak with solution called entropy viscosity solution. 4. In fully nonlinear PDE such as the Hamilton-Jacobi equation, there is a very different