shape calculus and free boundary problems

European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanm¨ aki, T. Rossi, S. Korotov, E. O˜ nate, J. P´ eriaux, and D. Kn¨ orzer (eds.) Jyv¨ askyl¨ a, 24–28 July 2004

SHAPE CALCULUS AND FREE BOUNDARY PROBLEMS Kari T. K¨ arkk¨ ainen? and Timo Tiihonen? ? Department

of Mathematical Information Technology P.O. Box 35 (Agora), FIN-40014 University of Jyv¨askyl¨a, Finland e-mails: [email protected], [email protected] web page: http://www.mit.jyu.fi/tiihonen/

Key words: Free boundary problems, shape sensitivity analysis, automatic differentiation. Abstract. In this paper we shall study different solution strategies for abstract form of free boundary problems. Using the tools of shape calculus we analyse different solution strategies. As a result of our analysis we provide guidelines for a generic approach for deriving efficient numerical algorithms for stationary FBP:s and discuss their implementation.

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Kari T. K¨arkk¨ainen and Timo Tiihonen

1

INTRODUCTION Let us consider following abstract free boundary problem (FBP), A(u, Ω) = 0, B(u, Ω) = 0.

(1) (2)

Here A corresponds to a well posed elliptic boundary value problem in domain Ω, and B respectively operates on the functions supported at the free boundary Σ ⊂ ∂Ω. It is supposed that function u can be solved from equation (1) for any given suitable domain Ω. The equations describing the free boundary are usually highly implicit with respect to the shape of the domain and can not be solved straightforwardly. In this paper we consider approaches where this system of equations is solved iteratively. We further restrict our attention to finding solutions with known topology. In the following we review briefly different solution strategies to solve the free boundary problems in the above form and formulate them to general principles. We shall formulate the solution strategies for the abstract problem (1)-(2) and for convenience of the reader then apply the strategies for the following model free boundary problem, so called Bernoulli free boundary problem: −∆u = 0 u=1 u=0 ∂u =λ ∂ν

in DΩ , on ∂D, on ∂Ω,

(3) (4) (5)

on ∂Ω,

(6)

¯ This equation models for example electro-chemical galvanization where DΩ = D \ Ω. or two dimensional inviscid irrotational fluid. For simplicity we assume here that the unknown free boundary is contained in a larger domain D. 2

SHAPE OPTIMIZATION APPROACH

In shape optimization approach one formulates a cost function that attains its minimum at a solution of the free boundary problem. Roughly speaking, optimization based approaches can be divided into two subclasses, first class assumes that the free boundary equations (1)–(2) are first order optimality conditions for some “energy” functional. Formally this means that there exists a function E(u, Ω) such that ∂E(u, Ω) ∂E(u, Ω) = 0 and = 0 ⇒ A(u, Ω) = 0 and B(u, Ω) = 0. ∂u ∂Ω However, this approach applies only for some restricted subclass of free boundary problems. Quite an extensive presentation of energy–based free boundary problems can be found in [6]. 2

Kari T. K¨arkk¨ainen and Timo Tiihonen

In the second class of shape optimization based approaches the additional free boundary condition is reformulated as a functional which attains its minimum at the solution of the free boundary problem. The other equation is considered as a constraint for this shape optimization problem. Latter we shall call this as least squares approach. For our abstract model problem this can read as min kB(u, Ω)k2 with subject to A(u, Ω) = 0, Ω∈O

(7)

where O is a suitable set of domains and k · k is some norm supported only on the free boundary. This formulation is much weaker than the original free boundary problem (1)– (2) since the second condition is relaxed and need not to be fulfilled exactly at the solution. Hence we can have a solution for the above shape optimization problem although we do not have solution for the free boundary problem. Therefore this formulation can be useful only from the numerical point of view. The least squares approach does not lead to unique formulation of the optimization problem. One can vary the norm in the cost function as well as the regularity of the set of admissible geometries. If the norms and regularities are not compatible one can either end up in minima where the cost function is strictly positive or to minimizing sequences that converge to fractal type of domains. Another important degree of freedom is how to write the free boundary equations as a state problem and an additional free boundary condition. It turns out that some choices are better from the point of view of the rate of convergence of the minimization algorithms. In the following sections we shall briefly apply shape optimization to our model free boundary problem. 2.1

Variational formulations

For general free boundary equations (1)–(2) we can not always construct an “energy” formulation. However, for our model problem it is well known that the solution of the model problem is a critical point of the functional [13], Z E(u) = ∇u · ∇u + λ2 I{u>0} dx, D

where I{u>0} is characteristic function of the set {u > 0}. This energy is minimized over all functions in space V := {u ∈ H 1 (D) | u = 1 on ∂D}. In this formulation the geometry is hidden into I{u>0} so that the geometry is not a variable in this definition. However, the characteristic function is nonsmooth thus producing troubles in minimization. To obtain a smooth formulation for the above problem we rewrite the energy functional as Z Z E1 (u, Ω) = ∇u · ∇u + λ2 dx. D

DΩ

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Kari T. K¨arkk¨ainen and Timo Tiihonen

Now Ω appears as an unknown variable in the energy. Thus we seek for the minimizer of E1 (u, Ω) in some space V × O. If the minimization is successful the minimizer describes the free boundary ∂Ω = {u = 0}. Alt and Caffarelli [1] showed for the first energy formulation that a nontrivial minimizer exists if and only if inf E(u) < E(0) = λ2 |D|. This is valid only if λ is larger than some critical value λc . 2.2

Least squares approach

Let us apply the least squares approach for our model problem. Now choosing as the cost function the L2 -norm at the solution on the free boundary, we end up with the following shape optimization problem: Z min J(Ω) = u2 dσ where u is such that u = 1 on ∂D and (8) Ω∈O ∂Ω Z Z (9) ∇u · ∇φ dx = λφ dσ for all φ ∈ H01 (D). DΩ

∂Ω

Clearly, if the FBP has a solution (u, Ω∗ ) and Ω∗ ∈ O, then u = 0 on ∂Ω∗ and therefore J(Ω∗ ) = 0. This formulation is by far not unique shape optimization formulation for our model problem. We could replace the L2 -norm by some other norm in the cost function. Or we could modify the state constraint, for example to the form, Z Z Z λφ dσ for all φ ∈ H01 (D). ∇u · ∇φ dx + αuφ dσ = DΩ

∂Ω

∂Ω

Here the coefficient α can be chosen freely without affecting the solution of the free boundary problem. However, α changes the conditioning of the Hessian of the cost function. It can be shown that α = H, the mean curvature of the free boundary, is in certain sense the optimal choice, [13]. The least squares approach can be applied even in the case when we impose the Dirichlet condition u = 0 for the state problem and formulate the cost function from ∂u − λ. This ∂ν has been studied by Haslinger et al, [8], who have found this approach especially suitable when the state problem is solved using fictitious domain method. 3

FIXED POINT APPROACH

In the fixed point approach to the free boundary problem the nonlinear problem is usually solved by constructing a sequence of linear problems. Fixed point algorithms do not commonly use gradient information but the algorithms can be based on the shape sensitivity analysis of continuous problem.

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Kari T. K¨arkk¨ainen and Timo Tiihonen

3.1

Trial methods

Trial methods can be characterized as Picard–type iterations for free boundary problems. Trial method to solve a free boundary problem of type (1)–(2) can be written in algorithmic form: 1. Set k = 0, choose initial domain Ω0 . 2. Solve uk from A(uk , Ωk ) = 0. 3. Construct Ωk+1 = F (uk , Ωk ), where F is chosen so that B(uk , Ωk+1 ) ≈ 0. 4. Update k = k + 1 and continue from step 2 until B(uk , Ωk ) is small enough. In particular, the above solution algorithm is generally simple to implement, but it is not always obvious to construct the updating step so that the method converges or so that the convergence is fast. One possible method to derive fixed point algorithms is to apply steepest descent method to the least squares formulation. If we apply this to our model problem (3)–(6) we can set up an algorithm as follows: 1. Set k = 0, choose initial domain Ω0 . = λ on the free boundary 2. Solve ∆uk = 0 with Neumann boundary condition αu+ ∂u ∂ν and uk = 1 on the fixed boundary ∂D. 3. Move free boundary with updating rule ∂Ωk+1 = ∂Ωk + βuk ν˜ .

4. Update k = k + 1 and continue from step 2 until uk is small enough. Here ν˜ is a smoothed normal vector field on the free boundary. This is required, since updating the boundary with unsmoothed normal vector field would decrease the smoothness of the free boundary. This algorithm has two parameters that have to be defined. The step length β and the tuning parameter α in the state problem. Rather obvious choice for the step length is β = 1/λ because it holds ∂u = λ on the free boundary at the solution. The optimal ∂ν tuning has been considered first by Garabedian [7] for our model problem. This was later generalized by Crank [3] for generic elliptic equation. In these approaches the driving goal was to find boundary conditions that minimize the role of boundary variations to the state solution. The optimal choice for our model problem is α = H, the mean curvature of the free boundary. The same solution was later found by Tiihonen [14] by studying the second order shape derivatives of the least squares formulation. 5

Kari T. K¨arkk¨ainen and Timo Tiihonen

Flucher and Rumpf [5] have analyzed the convergence of the algorithm for the model problem. Their analysis shows that the convergence suffers from the smoothing procedure so that the convergence is less than quadratic but still superlinear. For two dimensional case one can obtain the convergence rate 32 but in n–dimensional case only 65 can be achieved. We have also studied convergence in the presence of curvature term in the boundary condition in reference [11]. Detailed analysis in continuous case shows that the convergence factor is only 43 in two dimensional case. 3.2

Shape linearization

In shape linearization the nonlinear system of equations is solved by a Newton like method in functional spaces. The derivatives are calculated simultaneously with respect to the state u and geometry Ω. Newton iteration assumes that we have an initial guess for the solution (uk , Ωk ) and then we solve a linear equation to find new iterate (uk+1 , Ωk+1 ) close to the solution. In one Newton step we calculate the change δuk to the solution uk and change δΣk to the geometry Ωk , and then update (uk+1 , Ωk+1 ) = (uk , Ωk )+(δuk , δΣk ). Formally we obtain the system of equations A,u (uk , Ωk ) δuk + A,Σ (uk , Ωk ) δΣk = A(uk , Ωk ), B,u (uk , Ωk ) δuk + B,Σ (uk , Ωk ) δΣk = B(uk , Ωk ).

(10) (11)

Here ·,u means Fr´echet–derivative with respect to u and ·,Σ Fr´echet–derivative with respect to the geometry. There is no easy way to construct an appropriate function space for general geometries. The work by Zol´esio and Delfour [4] gives a detailed description of differentiation of functionals with respect to the geometry and, in addition, studies extensively different ways to construct valid topologies for geometries. Under appropriate smoothness assumptions the differentiation with respect to the geometry results to an operator that is supported only on the functions defined on the boundary of the domain that is differentiated. In particular, for our purpose the derivatives A,Σ and B,Σ are supported only on the free boundary. Thus the new unknown δΣk is also supported only on the free boundary. Let us now study the shape linearization for our model free boundary problem. The derivative with respect to state variable u reads as Z hA,u (u, Ω)δu, φi = ∇δu · ∇φ dx Ω

and the derivative with respect to geometry reads as Z hA,Σ (u, Ω)δΩ, φi = (∇Γ u · ∇Γ φ − λHφ) δΣ dσ. ∂Ω

Here H is the mean curvature of the free boundary. Above it was assumed that on ∂Ω in distributional sense. 6

∂φ ∂ν

=0

Kari T. K¨arkk¨ainen and Timo Tiihonen

The derivatives of B read as Z hB,u (u, Ω)δu, ψi =

δuψ dσ, ∂Ω

and



 ∂u hB,Σ (u, Ω)δΩ, ψi = ψ + Huψ δΣ dσ ∂ν ∂Ω As a result we end up with following iteration scheme: Given (uk , Ωk ) find (δuk , δΣk ) such that Z Z Z Z ∇δuk · ∇φ dx − (λHφ − ∇Γ uk · ∇Γ φ) δΣk dσ = ∇uk · ∇φ dx − λφ dσ, DΩk ∂Ωk DΩ ∂Ωk   Z Z Z ∂uk ψ + Huk ψ δΣk dσ = uk ψ, δuk ψ dσ + ∂ν ∂Ωk ∂Ωk ∂Ωk (12) for all (φ, ψ). The Newton step consists of updating the geometry by ∂Ωk+1 = ∂Ωk + ν˜ δΣk , ˜ k , where ˜· means smooth extension to the updated and updating the state by uk+1 = u˜k +δu geometry. The above system can be simplified further by taking into account the properties of the solution u at the solution of the free boundary. As it holds that u = 0 and ∂u = λ we ∂ν can substitute these to uk to get Z Z Z Z ∇δuk · ∇φ dx − λHφ δΣk dσ = ∇uk · ∇φ dx − λφ dσ, DΩk ∂Ωk DΩ ∂Ωk Z Z Z (13) δuk ψ dσ + λψ δΣk dσ = uk ψ, Z

∂Ωk

∂Ωk

∂Ωk

In above linearized system we can eliminate δΣk by setting δΣk = (uk − δuk )/λ. In particular, this results to the same algorithm that was presented in previous section as a superlinearly convergent algorithm. What makes the situation more interesting here is that the above derivation was done by just usual Newton–linearization and the resulting system of equations can be set up for general free boundary problem also. Thus we claim that the previous approach can be applied only to subclass of free boundary problems whereas this approach seems to be more general. However, quadratic convergence can not be expected because sufficient smoothness of the updates can not be obtained without smoothing. Moreover, usually it pays off to simplify some asymptotically vanishing terms from the equations defining the updating step. 4 4.1

IMPLEMENTATION Discretization of continuous algorithms

The algorithms introduced above have to be discretized to be useful in numerical solution of free boundary problems. In this respect the shape optimization problems of Section 2. are quite standard and do not require any special attention here. 7

Kari T. K¨arkk¨ainen and Timo Tiihonen

The algorithm of section 3.1 leads to a relatively standard state equation. The non trivial aspects are related to the approximation and smoothing of the normal field and the mean curvature on a piecewise polynomial boundary. Projection to piecewise linear functions is often adequate here. The main drawback of this approach is that the derivation of the algorithm is based on shape sensitivity analysis and can not hence be easily automated. Moreover, the approach applies only to a limited set of free boundary problems. The approach of section 3.2 is more general. However, it is also based on shape sensitivity analysis (which is bit less demanding in this case). The resulting system of equations is not of standard type. Hence, the numerical tools should be versatile enough to handle unknowns and operators on the boundary. 4.2

Fully discrete approach and automatic differentiation

Let us now analyse how the ideas presented above can be combined with the tools of automatic differentiation available for the discrete case. As such, fully discrete approaches for shape optimization or free boundary problems are known. However, they typically have the common feature that the finite element discretization inside the domain gets heavily involved in the solution process. That is, the dependency of the inner nodes on the boundary variations has to be resolved and taken into account when computing the sensitivities. This means additional work in compiling the state problems and also additional fill in in the corresponding linear equations. In what follows we study how to retain the relative ’compactness’ of the continuous approach and still exploit the ease of automatic differentiation. In the case of sensitivity analysis for a finite difference problem this has been studied by Borggaard et al [2], who proposed to modify the functional that was to be differentiated. Here we shall modify the implementation of the finite element method so that the automatic differentiation can be applied straightforwardly to the generic form of free boundary problem. ¿From the numerical point of view, we apply the automatic differentiation to calculate the derivative of residual (R1 (q, α), R2 (q, α)) ∈ Rn×m , where m is degrees of freedom in the geometry design and n is degrees of freedom of state. Here q denotes the state solution and α = (α1 , . . . , αm ) stands for geometry design vector, which defines the geometry of the finite element mesh. One Newton step to find zero point of the residual consists now of solving a linear system      A11 A12 dq R1 (q, α) = (14) A21 A22 dα R2 (q, α) Here A11 is the derivative of residual R1 with respect to q, A12 is the derivative of residual R1 with respect to α and respectively A21 derivative of R2 with respect to q, A21 derivative of R2 with respect to α.

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Kari T. K¨arkk¨ainen and Timo Tiihonen

In practical computations of shape derivatives we apply the chain rule [9, 10, 12], i.e. ∂f (x(α)) ∂f (x(α)) ∂x = . ∂αi ∂x ∂αi Here ∂x/∂αi can be provided in advance for finite elements by calculating the dependence of each node point with respect to the geometry design variable vector α. This approach has, however, one drawback. We need to compute the derivative of each nodal point with respect to the design variable α. Thus in the worst case we could have a full m × n matrix (times space dimension), for which m is the length of α and n is the number of nodal points in the finite element mesh. In practice, the mapping between the design parameter and the internal nodes has to be chosen so that the support of the mapping is small enough to guarantee efficient computation. In our case, matrices A21 and A22 are supported only on the free boundary by construction. As for A12 the standard procedure of automatic differentiation creates contributions also for internal nodes that have no counterpart in the continuous formulation. Our aim is to tune the automatic differentiation so that A12 would be supported on the free boundary only. To find the right modification we need to study automatic differentiation of local integrals in isoparametric finite element approach [9]. In isoparametric approach the local integrals are transformed to integrals over the reference element. We shall study this for a single element at first. In this context we shall adopt the notations used in [9], b· indicates that the variable is defined on reference element. In the following Te is a single element, Tˆ is the reference element and Fe : Tˆ → Te is a one-to-one mapping of Tˆ onto Te . Then the integrals defined on Te can be computed by integrating over the reference element Z Z f (Fe (ξ)) |J|dξ, f (x) dx = Tˆ

Te

where |J| is the determinant of the Jacobian J of mapping Fe . Standard way to calculate local integrals in isoparametric finite element method is to use local matrices and vectors in computation. First define ! ∂ϕ bp ∂ϕ b1 . . . ∂ξ1 b := ∂ξ1 G , ∂ϕ bp ∂ϕ b1 . . . ∂ξ2 ∂ξ2 where ϕk are shape functions and ∂ ϕ bk /∂ξj are derivatives of the shape functions for p-noded Lagrangian finite element in reference element Tb. Fe is now given by Fe (ξ) =

p X

ϕ(ξ)X b i

i=1

9

for ξ ∈ Tb,

Kari T. K¨arkk¨ainen and Timo Tiihonen

where X is a vector of nodal points, 

 X11 X12 X22 X22    X =  .. ..  .  . .  Xp1 Xp2 Here Xi1 and Xi2 are the coordinates of ith node in dimension two. It holds that [9]: ϕ(x) = ϕ(F b e −1 (x)), ∇ϕ(x) = J

−1

−1

∇ξ ϕ(F b e (x)) and b J = GX

(15) (16) (17)

for x ∈ Te . For automatic shape differentiation of the local integrals we proceed as follows. First we construct a mapping from the design variables to the nodal coordinates that is supported only on the boundary nodes (i.e. only the boundary nodes are considered to move in the analysis). Then in the actual automatic differentiation we assume that the finite element basis functions (in the physical element) do not depend on the design mapping. That is, we assume ϕ0 = 0 instead of ϕ˙ = 0 that holds for standard isoparametric mappings. In practice this is achieved by setting ˙ d ϕ(ξ) = ϕ(F ˙ e (ξ)) = ∇x ϕ · x(ξ) ˙ in the routine that computes the AD-derivatives of the test functions. When ∇ϕ is ˙ = 0 which is valid for P1 -elements and reasonable apdifferentiated, we set simply ∇ϕ proximation for higher order elements. When we compute the sensitivities of the local integrals we observe that the boundary perturbation affects the equilibrium at all nodes of the outermost element layer. Hence the support of the linearized system would include some internal nodes also, contrary to the continuous case. This is in fact analogous to the continuous approach to sensitivity analysis where it is observed that simple minded formal differentiation of the weak form does not lead to the correct answer. The sensitivity system must be interpreted in the strong form which in turn has the implication that the normal derivative of the test function should not contribute to shape sensitivity computations. The same observation should be implemented also in the discrete case. We study the shape sensitivity of the integral Z f (ϕi , ϕj , ∇ϕi , ∇ϕj ) dx. Kij = Te

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Kari T. K¨arkk¨ainen and Timo Tiihonen

Clearly, some of the ϕj :s are related to internal nodes and the corresponding contributions to not have counterparts in the continuous analysis. We want to create a situation where only the test functions related to boundary nodes are taken into account and in addition the contribution from their normal derivatives is eliminated. For that we construct test functions for the boundary nodes that have vanishing normal derivatives. Effectively this means that the internal test functions are explicitly dependent on the boundary test functions. In practice one constructs a projection mapping the rows of the full local matrix to rows corresponding to the boundary nodes only. For boundary elements the above approach can not be applied directly as there are no internal test functions to be manipulated. A typical boundary integral is of the form Z X g( qi ϕi )ϕj dσ. ∂Te ∩Σ

i

In standard isoparametric approach ϕi and ϕj are constant with respect to the boundary change in the normal direction. The only contribution from the automatic differentiation is therefore the discrete variant of mean curvature that describes the area change of the boundary integral. This would be correct if we would compute material derivatives with respect to the state solution. Now, however, the unknown in the perturbation system is the shape derivative of the P i δΣϕj from the sensitivity analysis. solution which means that we need a term gu i qi ∂ϕ ∂ν One way to achieve this is to write the evaluation of boundary integrals in the same form as evaluation of volume integrals, the only difference being in the choice of the quadrature formula that now should be supported on the boundary. Above discussion of elimination of normal derivatives with projection mapping applies for boundary integrals corresponding matrix block A12 . For the computation of matrix block A22 we should have test function φj supported only on the boundary so that there would not appear any contribution of normal derivative of φj . However, if the test function φj is not restricted on the boundary P ∂ϕ we get extra term g( i qi ϕi ) ∂νj δΣ to the sensitivity equation. In our implementation this term was subtracted explicitly. In the case where the boundary elements are managed individually, without direct relationship with the corresponding volume elements, one can construct an artificial volume element for each boundary element. This is illustrated figure 1 for artificial Q1 elements defined for the boundary segments of the P1 -mesh. This however necessitates the interpolation of the solution from P1 –mesh to Q1 –layer. In our numerical computations we shall use only Q1 –elements. As a conclusion we need following modifications for standard finite element program: • The shape functions are modified so that the discrete finite element function space does not depend on the design parameter. • The shape sensitivities are calculated only elements near the free boundary and for the free boundary integrals. 11

Kari T. K¨arkk¨ainen and Timo Tiihonen

Figure 1: Q1 –element layer generated for P1 –element mesh.

• Normal derivatives of the test functions are subtracted by projection for the area integrals and by subtracting the normal derivative part for boundary integrals. 5

NUMERICAL RESULTS

In this section we shall apply the discrete shape linearization method for our model problem in order to evaluate the discrete method by comparing it to shape linearized equations in the continuous case. We study the convergence of our model problem in star-shaped domain. w The finite element program that was modified was Numerrin 2.0 [12]. In this finite element library automatic differentiation is implemented already but for our case the kernel had to be modified. A extra boundary value problem was solved at each iterate to continuate the geometry update from the free boundary to the inner part of the finite element mesh. The tests were performed on a Linux-workstation with AMD Athlon CPU 1.4 Ghz processor with 256Mb memory. 5.1

Model problem

Our aim was to set up a test case which shows that discrete shape linearization works on our model problem. The geometry of the domain can be seen in figure 2. In the Neumann condition (5) we used value λ = 1 and we changed the free boundary to be outer boundary, and fixed boundary to be inner boundary of the geometry. We further used zero Dirichlet value for fixed boundary and u = 1 on the free boundary. The fixed geometry consists of four segments of circle with radius 0.316, corner points of the fixed boundary are located at (−0.4, 0.0), (0.0, 0.2), (0.4, 0.0) and (0.0, −0.2). We tested the convergence of the algorithm with different meshes and compared the results to implementations of continuous algorithm forms (12) (denoted CSL) and (13) (denoted ISL). Performance times of different algorithms are shown in table 1. We observed that discrete shape linearization method (denoted DSL) is slower due to the automatic differentiation of variables with respect to the geometry, but the slow down is only 12

Kari T. K¨arkk¨ainen and Timo Tiihonen

Figure 2: Initial 128 × 16 mesh (left) and final mesh (right) for model problem.

approximately 6% from the continuous form of the algorithms. However, as the convergence slows down compared to the continuous algorithms the performance of the discrete shape linearization algorithm is weak compared to continuous form of the algorithm, see table 1. DSL CSL ISL

64 × 8 128 × 16 256 × 32 4.28 14.0 63.7 3.84 14.0 63.9 3.81 14.8 67.7

Table 1: Execution times of the algorithms.

In figure 3 there are plotted the sum of L2 -norms of updates. The L2 -norm for the geometry update is calculated on the free boundary. We can observe that the convergence of the implicit shape linearization method (ISL) accelerates as the mesh is refined. This can not be observed from the other two algorithms. 6

CONCLUSIONS

The shape calculus provides tools for systematic study and construction of algorithms for solving stationary free boundary problems. In the general case the optimal linearization of the state problem lead to coupled systems with additional unknown on the free boundary. An important result is that the tools of automatic differentiation can be tuned so that they can be used systematically in construction of efficient iterative methods for discretised free boundary problems.

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Kari T. K¨arkk¨ainen and Timo Tiihonen

CSL ISL DSL

1

0.0001

0.0001

1e-08

1e-08

1e-12

0

2

4

6

8

DSL DSL DSL ISL ISL ISL

1

1e-12

10

0

2

4

6

64x8 128x16 256x32 64x8 128x16 256x32

8

10

Figure 3: Convergence of different shape linearization methods for 256 × 32 mesh (left) and convergence of discrete shape linearization method and implicit shape linearization method as mesh is refined (right) for model problem.

REFERENCES [1] H. Alt and L. Caffarelli. Existence and regularity for a minimum problem with free boudary. J. Reine Angew. Math., 325:105–144, 1981. [2] J. Borggaard and A. Verma. On efficient solutions to the continuous sensitivity equation using automatic differentiation. SIAM J. Sci. Comput., 22(1):39–62 (electronic), 2000. [3] J. Crank. Free and moving boundary problems. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1984. [4] M. C. Delfour and J.-P. Zol´esio. Shapes and geometries. Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Analysis, differential calculus, and optimization. [5] M. Flucher and M. Rumpf. Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math., 486:165–204, 1997. [6] A. Friedman. Variational Principles and Free-Boundary Problems. Wiley, 1982. [7] P. Garabedian. The mathematical theory of three dimensional cavities and jets. Bull Amer. Math. Soc., 62:219–235, 1956. [8] J. Haslinger, T. Kozubek, K. Kunisch, and P. G. Shape optimization and fictitious domain approach for solving free boundary value problems of bernoulli type. Computational Optimization and Applications, 26:231–251, 2003. [9] J. Haslinger and R. A. E. M¨akinen. Introduction to shape optimization. Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. Theory, approximation, and computation. 14

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[10] K. Hiltunen, L. Mika, N. Antti, and T. Pasi. Using mathematical concepts in software design of computational mechanics. In Proceedings of the VIII Finnish Mechanics Days, 2003. [11] K. K¨arkk¨ainen. Bernoulli’s free boundary problem with curvature dependent boundary conditions. In Free Boundary Problems: Theory and Applications, volume 14 of GAKUTO International Series, pages 239–254. Gakkotosho, Tokyo, 2000. [12] Numerola Oy. Numerrin 2.0 k¨aytt¨ aj¨ an opas, 2003. (in finnish). [13] T. Tiihonen. Shape optimization and trial methods for free boundary problems. RAIRO Mod´el. Math. Anal. Num´er., 31(7):805–825, 1997. [14] T. Tiihonen and J. J¨arvinen. On fixed point (trial) methods for free boundary problems. In S. Antontsev, K.-H. Hoffmann, and A. Khludnev, editors, Free boundary problems in continuum mechanics. Birkh¨auser–Verlag, 1991.

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