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Fictitious Domain Approach for Spectral/Hp Element Method Copyright c 2007 Tech Science Press CMES, vol.17, no.2, pp.95-114, 2007 Fictitious Domain Approach for Spectral/hp Element Method L. Parussini1 Abstract: We propose a fictitious domain approaches have been developed to implement method combined with spectral/hp elements for immersed conditions and we can find different the solution of second-order differential prob- applications: to acoustics [Heikkola, Kuznetsov, lems. This paper presents the formulation, and Lipnikov (1999); Hetmaniuk and Farhat validation and application of fictitiuos domain- (2002); Hetmaniuk and Farhat (2003)], fluid dy- spectral/hp element algorithm to one- and two- namics [Glowinski, Pan, and Periaux (1994); dimensional Poisson problems. Fictitious domain Glowinski, Pan, and Periaux (1995); Glowin- methods allow problems formulated on an intri- ski, Pan, and Periaux (1997); Glowinski, Pan, cate domain Ω to be solved on a simpler domain Hesla, Joseph, and Periaux (1999); Glowinski, Π containing Ω. The Poisson equation, extended Pan, Hesla, Joseph, and Periaux (2001)], bio- to the new domain Π, is expressed as an equiv- medical problems [Arthurs, Moore, Peskin, Pit- alent set of first-order equations by introducing man, and Layton (1998); Roma, Peskin, and the gradient as an additional indipendent variable, Berger (1999); Jung and Peskin (2001); de Hart, and spectral/hp element method is used to develop Peters, Schreurs, and Baaijens (2000); Baaijens the discrete model. Convergence of relative en- (2001); de Hart, Peters, Schreurs, and Baaijens ergy norm η is verified computing smooth solu- (2003)]. In general we have to distinguish the tions to one- and two-dimensional Poisson equa- case of Dirichlet, Neumann or Robin conditions. tions. Thermal field calculations for heatsink pro- Methods to implement Dirichlet conditions can file is presented to demonstrate the predictive ca- be classified in penalty methods and Lagrangian pability of the proposed formulation. methods, as illustrated by [Heikkola, Kuznetsov, Neittanmaki, and Pironneau (2003)]. There are Keyword: fictitious domain, Lagrange multi- examples of Elimination method [Zienkiewicz pliers, spectral/hp element method, Poisson prob- and Taylor (2000)], Penalty method [Zienkiewicz lem. and Taylor (2000); Ramiere, Angot, and Bel- liard (2005)], Distributed Lagrangian method 1 Introduction [Haslinger, Maitre, and Tomas (2001); Glowinski, Pan, Hesla, and Joseph (1999); Glowinski, Pan, The main motivation for fictitious domain ap- Hesla, Joseph, and Periaux (2000)], Boundary proach, alternatively called immersed boundary Lagrangian method [Stenberg (1995); Joly and method, is that, defining the extended problem Rhaouti (1999)], Fat Boundary method [Maury on a simple domain, enables the use of effi- (2001)]. Among all the possible approaches, a cient discretization methods on simple structured technique which is popular, given its efficiency, grids. Literature on fictitious domain methods is to enforce the Dirichlet condition by Lagrange goes back to the sixties [Saul’ev (1963)]. In multipliers, which is the method we propose in the following years there has been a crescent in- this paper. Unfortunately, the same approach can terest on this kind of methodologies [Babuska not be adopted for Neumann immersed boundary (1973); Pitkaranta (1979); Pitkaranta (1980); condition because that one can not be directly en- Pitkaranta (1981); Barbosa and Hughes (1991); forced by Lagrange multipliers [Hetmaniuk and Barbosa and Hughes (1992)]. So that several Farhat (2003)]. Enforcing Neumann boundary conditions within a fictitious domain framework 1 University of Trieste, Mech. Eng. Dept., ITALY. 96 Copyright c 2007 Tech Science Press CMES, vol.17, no.2, pp.95-114, 2007 is not an easy task. To treat them within a as follows: fictitious domain framework least-squares for- find φ(x) such that mulations have been proposed by [Dean, Dihn, Glowinski, He, Pan, and Periaux (1992)], vector −φ = f inΩ (1) s dual formulations with Lagrange multipliers have φ = φ on Γφ (2) been presented in [Joly and Rhaouti (1999); Quar- ∇φ • nˆ = qs on Γ (3) teroni and Valli (1999)] and an algebraic fictitious n q domain approach in [Rossi and Toivanen (1999); where Γ = Γφ Γq and Γφ Γq = 0,/ f is the Makinen, Rossi, and Toivanen (2000)]. We have source term, nˆ is the outward unit normal on chosen to impose it via Lagrange multipliers, in- boundary Γ, φ s is the prescribed value of φ on troducing the gradient of the primal variable as Γ s boundary φ and qn is the prescribed normal flux an unknown and converting Neumann condition on boundary Γq. into a Dirichlet-like condition for this new vari- To enable the implementation of Neumann able. This technique seems to be very efficient, at boundary condition via Lagrange multipliers, we the expense of an increase of computational cost proceed by replacing the Poisson problem, Eq. 1- as it replaces a scalar problem by a vector prob- Eq. 3, with its equivalent first-order system, at the lem. expense of additional variables. The equivalent New approach, we present in this paper, is the problem writes as: coupling of fictitious domain together with an find φ(x) and u(x) such that high order method. To discretize the problem un- der study we use the spectral/hp element method, u−∇φ = 0inΩ (4) based on higher order functions, locally defined −∇• u = f inΩ (5) over finite size parts of domain. The advantage φ = φ s Γ of such kind of method, in comparison with tra- on φ (6) • = s Γ ditional finite element method, is its exponential u nˆ qn on q (7) convergence property with the increasing of poly- nomial order p. Moreover using fictitious domain where u is a vector valued function whose com- φ approach, where extended problem is defined on ponents are the fluxes of scalar function ,asde- a simple domain, enables the use of efficient com- fined in Eq. 4. putational grids, in our case just simple Cartesian We can write the system Eq. 4-Eq. 5 defined on Ω grids. domain in matrix form as: Good accuracy properties of the method are LΦ = F (8) demonstrated by numerical experiments. These ones are performed with several mesh size and where L is the matrix operator, Φ is the vector of polinomyal order of modal function to better unknowns and F is the vector of known terms. In quantify the performance of the proposed solution two dimensions it is: ⎡ ⎤ procedure. Finally, the calculation of the thermal ∂ ⎧ ⎫ ⎧ ⎫ 10− ⎨ φ ⎬ ⎨ ⎬ field for a heatsink fin illustrates the capabilities ⎢ ∂x ⎥ x 0 ⎣ 01− ∂ ⎦ φ = of the method in a practical engineering example. ∂y ⎩ y ⎭ ⎩ 0 ⎭ (9) − ∂ − ∂ φ f ∂x ∂y 0 2 Formulation of the problem: the Poisson φ = ∂φ φ = ∂φ equation with x ∂x y ∂y . Let Ω be the closure of an open bounded region 3 Fictitious domain method Ω in ℜn,wheren represents the number of space =( ,..., ) dimensions, and let x x1 xn be a point in Several variants of fictitious domain method ex- Ω = Ω ∂Ω,where∂Ω = Γ is the boundary of ist: the basic idea is to extend the operator and the Ω. We want to solve the Poisson problem stated domain into a larger simple shaped domain. The Fictitious Domain Approach for Spectral/hp ElementMethod 97 2 most important ways to do this are algebraic and Π ⊃ Ω with fΠ extension of f to L (Π) and im- functional analytic approaches. In algebraic fic- mersed constraints enforced via Lagrange mul- titious domain methods the problem is extended tipliers. The variational formulation of the new typically at algebraic level in such a way that the equivalent problem will be: solution of original problem is obtained directly as a restriction of the solution of extended prob- , − ,∇φ + ,λ = lem without any additional constraint. There are v u Π v Π v q Γq 0 (12) several variants of such an approach [Rossi and −ϕ,∇• uΠ +ϕ,λφ Γφ =(ϕ, f )Π (13) Toivanen (1999); Makinen, Rossi, and Toivanen μ ,uΓ = 0 (14) (2000)] and they can be rather efficient, but typi- q q μ ,φ = cally they are restricted to quite a narrow class of φ Γφ 0 (15) problems. where Γ = ∂Ω and Γ = Γφ Γ . The Lagrange More flexibility and better efficiency can be ob- q multiplier defined on Γφ is denoted by λφ , with μφ tained by using a functional analytic approach the associated weight function, and the Lagrange where the use of constraints ensures that the so- multiplier defined on Γ is denoted by λ , with μ lution of extended problem coincides with the so- q q q the associated weight function (see [Zienkiewicz lution of original problem. In our implementation and Taylor (2000)] for more details). we enforce constraints by Lagrange multipliers. The solution of Poisson problem, Eq. 1-Eq. 3, will For simplicity, let us consider the model problem be the restriction to Ω of the solution of Eq. 12- Eq. 4-Eq. 5 with homogeneous boundary condi- Eq. 15, which are defined on domain Π. tions. The standard approach to solve it is to search for the solution which minimizes the vari- 4 Discretization: spectral/hp element method ational problem, in practice: find (φ,u) ∈ X such that for all admissible weight- The problem Eq. 12-Eq. 15 can not be solved ing functions (ϕ,v) ∈ X analytically and therefore it is necessary to use a numerical method to get approximated solu- tion. The spectral/hp element method is a numeri- , − ,∇φ = v u Ω v Ω 0 (10) cal technique for solving partial differential equa- −ϕ,∇• uΩ =(ϕ, f )Ω (11) tions based on variational formulation of bound- ary and initial value problems [Karniadakis and where ·,·Ω and (·,·)Ω denote the stand- = Sherwin (1999); Pontaza and Reddy (2003); Es- ard H1 and L2 inner products and X 1 1 kilsson and Sherwin (2005); Gerritsma and Maer- (φ,u) ∈ H (Ω) ×H (Ω) : φ|Γ = 0, u• nˆ|Γ = 0 .
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