Non-Incremental Boundary Element Discretization of non-linear heat equation based on the use of the Proper Generalized Decompositions

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Non-Incremental Boundary Element Discretization of

non-linear heat equation based on the use of the Proper

Generalized Decompositions

Gaël Bonithon, Pierre Joyot, Francisco Chinesta, Pierre Villon

To cite this version:

Gaël Bonithon, Pierre Joyot, Francisco Chinesta, Pierre Villon. Non-Incremental Boundary Element Discretization of non-linear heat equation based on the use of the Proper Generalized Decomposi-tions. 11th International Conference on Advances in Boundary Element Techniques, Jul 2010, Berlin, Germany. �hal-01580912�

Non-Incremental Boundary Element Discretization of non-linear heat

equation based on the use of the Proper Generalized Decompositions

G. Bonithon

1,4

, P. Joyot

1

, F. Chinesta

2

and P. Villon

3 1 _{ESTIA-Recherche, technopole izarbel, 64210 Bidart, France, pjoyot@estia.fr}

2 _{EADS Corporate Foundation International Chair, GEM CNRS-ECN, 1 rue de la Noë BP 92101,}

44321 Nantes cedex 3, France, Francisco.Chinesta@ec-nantes.fr

3_{UTC-Roberval UMR 6253, 60200 Compiègne, France, pierre.villon@utc.fr} 4 _{EPSILON Ingénierie, 10 rue Jean Bart, BP 97431, France, 31674 Labège Cedex}

Keywords: Boundary element method, Separated representations, Proper Generalized Decomposition

Abstract. In this work, we propose a new approach for solving the heat equation within the Boundary Elements method framework. This technique lies in the use of a separated representation of the unknown field that allows decoupling the space problem (that results steady state) from the temporal one (one dimensional that only involves the time coordinate).

Introduction

The Boundary Elements Method (BEM) allows efficient solution of partial differential equations whose kernel functions are known. The heat equation is one of these candidates when the thermal parameters are assumed constant (linear model). When the model involves large physical domains and time simulation intervals the amount of information that must be stored increases significantly. We propose here an alternative strategy able to change the nature of the problem. Thus, the temper-ature field involved by the so called heat equation is approximated using a separated representation involving products of space and time functions. This kind of approximation is not new, in fact, proper orthogonal decomposition [1] allows such one decomposition, but in this case this decomposition must be applied a posteriori, i.e. on the transient solution of the considered model.

The technique that we propose in this paper allows to transform the transient model in a sequence of space problems (all of them steady state) and time problems (that only involve the time coordinate). This iteration procedure leads to a proper space-time generalized decomposition of the model solu-tion. The efficiency of such one approach was proven in [2, 3, 4]. In our knowledge, this technique has never been coupled with a BEM for solving the resulting steady problem defined in the physical domain.

We start summarizing the main ideas of the Proper Generalized Decomposition and we will focus on the application of such technique in the context of the BEM. Finally, numerical example, with a non linear source term, will be presented and discussed.

A Proper Generalized Decomposition Boundary Element Method Let us consider the heat equation

∂u

∂t − aΔu = f (u) inΩ × (0,Tmax] (1) with homogeneous initial and boundary conditions, where a is the diffusion coefficient,Ω ⊂ Rd,d ≥ 1, Tmax > 0. The aim of the separated representation method is to compute N couples of functions

{(Xi,Ti)}i=1,...,N such that{Xi}i=1,...,N and{Ti}i=1,...,N are defined respectively inΩ and (0,Tmax] and the solution u of this problem can be written in the separate form

u(x,t) ≈ N

∑

i=1

Ti(t) · Xi(x) (2)

The weak formulation yields: Find u(x,t) such that

 Tmax 0  Ωu ∂u ∂t− aΔu − f (x,t)  dx dt = 0 (3)

for all the functions u(x,t) in an appropriate functional space.

We compute now the functions involved in the sum (eq (2)). We suppose that the set of functional couples {(Xi,Ti)}i=1,...,n with 0≤ n < N are already known (they have been previously computed) and that at the present iteration we search the enrichment couple(R(t),S(x)) by applying an alternat-ing directions fixed point algorithm that after convergence will constitute the next functional couple (Xn+1,Tn+1). Hence, at the present iteration, n, we assume the separated representation

u(x,t) ≈ n

∑

i=1

Ti(t) · Xi(x) + R(t) · S(x) (4)

The weighting function uis then assumed as

u= S · R+ R · S (5)

Introducing (eq (4)) and (eq (5)) into (eq (3)) it results

 Tmax 0  Ω(S · R _{+ R · S}_{) ·}_S·∂R ∂t − aΔS · R  dx dt = = Tmax 0  Ω(S · R _{+ R · S}_{) ·}  f(x,t) − n

∑

i=1 Xi·∂T i ∂t + a n

∑

i=1 ΔXi· Ti  dx dt (6)

We apply an alternating directions fixed point algorithm to compute the couple of functions(R,S): • Computing the function S(x).

First, we suppose that R is known, implying that Rvanishes in (eq (5)). Thus, eq (6) writes

 ΩS _{· (α} tS− aβtΔS) dx =  ΩS _·  γt(x) − n

∑

i=1 αi tXi+ a n

∑

i=1 βi tΔXi  dx (7)

where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ αt=  Tmax 0 R(t) · ∂R ∂t(t)dt αi t =  Tmax 0 R(t) · ∂Ti ∂t(t)dt βt =  Tmax 0 R 2_(t)dt βi t =  Tmax 0 R(t) · Ti(t)dt γt(x) =  Tmax 0 R(t) · f (x,t) dt; ∀x ∈ Ω (8)

The weak formulation (eq (7)) is satisfied for all S, therefore we could come back to the asso-ciated strong formulation

αtS− aβtΔS = γt− n

∑

i=1 αi tXi+ a n

∑

i=1 βi tΔXi (9)

that one could solve by using any appropriate discretization technique for computing the space function S(x).

• Computing the function R(t).

From the function S(x) just computed, we search R(t). In this case S vanishes in (eq (5)) and eq (6) reduces to  Tmax 0  Ω(S · R _{) ·}_S·∂R ∂t − aΔS · R  dx dt = = Tmax 0  Ω(S · R _{) ·}  f(x,t) − n

∑

i=1 Xi·∂T i ∂t + a n

∑

i=1 ΔXi· Ti  dx dt (10)

where all the spatial functions can be integrated inΩ. Thus, by using the following notations ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ αx =  ΩS(x) · ΔS(x)dx αi x =  ΩS(x) · ΔXi(x)dx βx=  ΩS 2_(x)dx βi x=  ΩS(x) · Xi(x)dx γx(t) =  ΩS(x) · f (x,t) dx; ∀t (11)

equation (eq (10)) reads

 Tmax 0 R·  βx∂R ∂t − aαxR− γx(t) + n

∑

i=1 βi x ∂Ti ∂t − n

∑

i=1 aα_xi· Ti  dt= 0 (12) As eq (12) holds for all S, we could come back to the strong formulation

βx∂R ∂t = a · αx· R + γx(t) − n

∑

i=1 βi x· ∂Ti ∂t + n

∑

i=1 a· α_xi· Ti (13) which is a first order ordinary differential equation that can be solved easily (even for extremely small times steps) from its initial condition.

These two steps must be repeated until convergence, that is, until verifying that both functions reach a fixed point.

The BEM is used to solve eq (9). We can notice that this equation defines a steady state elliptic equation with constant coefficients.

Numerical example

We considered a simple rectangular domain Ω = (0,1) × (0,1) and a time interval I = (0,1]. The source term is set to f(u) = u2(1 − u), the boundary conditions and the initial condition is set to an exact solution of this problem given by :

ure f(x,t) = e η(x,t) 2+ eη(x,t) withη (x,t) = √1 2  x+√t 2 

The domain boundaryΓ consists of n_Γ× n_Γ segmentsΓi. The time intervalI is discretized by using n_τ nodes, uniformly distributed.

First we are analyzing the convergence rates as a function of the space discretization (i.e. n_Γ). For all the space meshes the time discretization (i.e. n_τ) is adapted in order to reach the maximum precision. Figure 1 show the evolution of the L2error in time and space as a function of the level of approxima-tion, that is, as a function of the number of functional couples Xi(x) · Ti(t) involved in the approxima-tion of u(x,t) for different meshes. This error is defined by:

en= n

∑

i=1 Xi(x) · Ti(t) − ure f(x,t) L2 Ω×I ure f_(x,t) L2 Ω×I

We can notice that for a given number of functional couples the error endecreases when nΓincreases, reaching an asymptotic value. For reducing the value of the error we must increase n_Γ as well as the number of functional couples Xi(x) ·Ti(t) involved in the functional approximation. In the case of the example here addressed we must consider 4 functional couples for reaching a quatratic convergence rate for 4 n_Γ 16.

Figure 3 depicts functions{X₁(x),T₁(t)}, {X₂(x),T₂(t)}, {X₃(x),T₃(t)} for n_Γ= 16 and n_τ = 256. Finally, figure 2 depicts the unknown field u(x,t).

Conclusion

1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 log₁₀(1/n_Γ) 5.5 5.0 4.5 4.0 3.5 3.0 log 10 (en ) 2 nRS=1 n_RS=2 nRS=3 nRS=4

Fig. 1: Evolution of the error en versus the space discretization for different levels of approximation n.

and some ordinary differential equations that only involve the time coordinate. Significant reduction of CPU time is expected due to the non-incremental nature of the proposed technique, as well as a significant reduction of the amount of information to be stored. As shown by the numerical exemple, this technique seems specially well adapted for the treatment of non-linear transient BEM models.

[

min

=0

.

68

, max

=1

.

00] × 5

.

3

e

−01 0.700 0.750 0.800 0.850 0.900 0.950 1 000 1.000

t

=2

.

4

e

−01 [

min

=0

.

69

, max

=1

.

00] × 5

.

7

e

−01 0.700 0.750 0.800 0.850 0.900 0.950 1 000 1.000

t

=5

.

0

e

−01 [

min

=0

.

71

, max

=1

.

00] × 6

.

0

e

−01 0.750 0.800 0.850 0.900 0.950 1 000 1.000

t

=7

.

6

e

−01 [

min

=0

.

72

, max

=1

.

00] × 6

.

3

e

−01 0.760 0.800 0.840 0.880 0.920 0.960 1 000 1.000

t

=1

.

0

e

+00

-0.900 -0.750 -0.600 -0.450 -0.300 -0.150 X1(x) 0 1.0 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2×1.7e−03 T1(t) -0.200 0.000 0.200 0.400 0.600 0.800 X₂(x) 0 1.0 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2×6.7e−04 T2(t) -0.900 -0.750 -0.600 -0.450 -0.300 -0.150 0.000 0.000 X₃(x) 0 1.0 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6×5.5e−05 T3(t)

References

[1] F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, and F. Boust. Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization. Computer Meth-ods in Applied Mechanics and Engineering, 197(5):400 – 413, 2008.

[2] A. Ammar, F. Chinesta, and P. Joyot. The nanometric and micrometric scales of the structure and mechanics of materials revisited: An introduction to the challenges of fully deterministic numeri-cal descriptions. International Journal for Multisnumeri-cale Computational Engineering, 6(3):191–213, 2008.

[3] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Journal of Non-Newtonian Fluid Mechanics, 139(3):153 – 176, 2006.

[4] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of com-plex fluids: Part ii: Transient simulation using space-time separated representations. Journal of Non-Newtonian Fluid Mechanics, 144(2-3):98 – 121, 2007.