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On the use of model reduction in material and processes
simulation: a way for realizing simulation dreams
Francisco Chinesta, Amine Ammar, Elías Cueto
To cite this version:
Francisco Chinesta, Amine Ammar, Elías Cueto. On the use of model reduction in material and processes simulation: a way for realizing simulation dreams. [Research Report] GEM. 2011. �hal-01634604�
ON THE USE OF MODEL REDUCTION IN MATERIAL AND PROCESSES SIMULATION: A WAY FOR REALIZING SIMULATION DREAMS
F. Chinesta1, A. Ammar2, E. Cueto3
1
EADS Corporate Foundation International Chair, GeM CNRS - Centrale Nantes 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France
Francisco.Chinesta@ec-nantes.fr
2
Laboratoire de Rhéologie, CNRS – INPG – UJF, 1301 rue de la piscine, BP 53, Domaine universitaire, F-38041 Grenoble cedex 9, France
Amine.Ammar@ujf-grenoble.fr
3
I3A, Universidad de Zaragoza, María de Luna, 7, E-50018 Zaragoza, Spain.
ecueto@unizar.es ABSTRACT
In this work we analyze the possibilities of applying model reduction in the numerical modelling of forming processes. The use of such strategies allows impressive computing time savings in the numerical simulations of complex models without degrading the solution accuracy. For this purpose we apply proper generalized decompositions of multidimensional and/or non-incremental transient models that can be associated to usual computational physics and mechanics models.
KEYWORDS: Numerical strategies, Model reduction, PGD, Material Forming 1. INTRODUCTION
The fine description of the mechanics and structure of materials at the micro, nano and sub-nanometric scales introduces some specific challenges related to the impressive number of degrees of freedom required or to the highly dimensional spaces in which those models are defined. Moreover, usual models encountered in computational physics and mechanics can be transformed in multi-dimensional models allowing for very general solutions as we describe later in section 2. Despite the fact that spectacular progresses have been accomplished in the context of computational mechanics in the last decade, the efficient treatment of those multi-dimensional models, needs further developments. Moreover
The brute force approach cannot be considered as a possibility for treating this kind of models. We can understand the catastrophe of dimension by assuming a model defined in a hyper-cube Ω of dimension D: Ω = −
]
L L,[
D. Now, if we define a grid to discretize the model, as it is usually performed in the vast majority of numerical methods (finite differences, finite elements, finite volumes, spectral methods etc.), consisting of M nodes on each direction, the total number of nodes will be M . If we assume that for example D M =10 (an extremely coarse description) and D=80 (much lower than the usual dimensions required in quantum or statistical mechanics) the number of nodes in Ω reaches the astronomical value of 10 that 80 represents the presumed number of elementary particles in the universe!We come back to the practical interest of multi-dimensional models later. In what follows in the present section we are revisiting a technique able to circumvent the curse of dimensionality issue.
1.1 Multidimensional solvers based on proper generalized decompositions
We start writing the polynomial approximation of a generic multi-dimensional function
(
1, 2, , D)
u x x ⋯ x in the whole domain as:
( )
1( )
1( )
( )
1 1 1 k D i N i N i i i D D k k i i k u X x X x X x = = = = = = ≈∑
=∑
∏
x ⋯ (1)The coordinate xi is not necessarily one-dimensional, but in any case it is defined in a space of moderate dimensions (1D, 2D or 3D), i.e. xi∈Ωi, di, 3
i ï
x ⊂ ℜ d ≤ . The model results then
defined in the whole domain 1
1 D d d D + + Ω = Ω × ×Ω ⊂ ℜ ⋯
⋯ . One of this coordinates could be the
time involved in transient models.
It is also well known that several model solutions can be approximated by a finite, and sometimes quite reduced number of functional products. Expression (1) involves N× ×M D
degrees of freedom instead of the MD required in mesh-based discretization techniques.
In that follows we are describing a new advanced technique that combines a separated representation and an adaptation procedure able to build up gradually each product of functions involved in (1) until reaching the convergence. It has some resemblances with the functional approximation used within the LATIN framework, the radial approximation making use of a space-time separated representation (see [8] and the references therein) as well as with to the ones employed in the post-Hartree-Fock methods [4]. This technique has been successfully applied in a variety of linear, non linear, stationary and transient problems [1] [2] [5] [9]. In what follow we are revisiting the main ideas of such decomposition technique. For the sake of simplicity we are considering a simple multi-dimensional diffusion problem in D dimensions:
( )
(
)
] [
(
)
2 1 , , , 0, 0 D T D u f x x L u ∇ = = ∈Ω = ∈∂Ω = x x x ⋯ (2)where the general form of the right term is given by
( )
1( )
1( )
( )
1 1 1 k D i m i m i i i D D k k i i k f F x F x F x = = = = = = ≈∑
=∑
∏
x ⋯ (3)decomposition that can be performed by using singular value decomposition.
The iteration scheme used to build up the solution (1) proceeds performing an enrichment of the approximation basis at each iteration. Thus, knowing the approximation at iteration the n:
( )
1( )
1( )
( )
1 1 1 k D i n i n n i i i D D k k i i k u X x X x X x = = = = = = =∑
=∑
∏
x ⋯ (4)the approximation basis could be enriched by adding a new product of functions 1
( )
1 k D n k k k X x = + =
∏
( )
( )
( )
1 1 1 k D n n n k k k u u X x = + + = = +∏
x x (5)that needs for the determination of the D involved functions Xkn 1
( )
xk+
. For this purpose the trial function:
( )
( )
( )
1 1 1 k D k D i n i k k k k i k k u X x R x = = = = = = =∑
∏
+∏
x (6)is injected in the weak formulation, where Rk
( )
xk Xkn 1( )
xk+
≡ are the unknown fields of the non linear system obtained, whose size is D M× . The associated test functions are taken, again in the Galerkin’s framework, as:
( )
( )
( )
* * 1 1 j D k D j j k k j k k j u R x R x = = = = ≠ = ∑
∏
x (7)Now, as soon as the functions Rk
( )
xk have been determined, the searched functions( )
1
n k k
X + x are obtained by identifying these functions with the convergedRk
( )
xk functions. This algorithm has been successfully used to solve models involving hundred dimensions needing of the order of ~10300 degrees of freedom if one proceeds in the finite element framework. The construction of the separated solution only needed of around 20 minutes using Matlab on a standard personal computer!. The separate representation considered in (1) only needs approximations defined in spaces of moderate dimensions di and then integrations in such moderate dimensional spaces because the integral of a product of functions in a hyper-domain can be written as the product of the integrals defined in the domains Ωi.1.2 Illustrating the PGD construction
In what follows we are illustrating the construction of the Proper Generalized Decomposition by considering a quite simple problem, the parametric heat transfer equation:
0 u k u f t ∂ − ∆ − = ∂ (8)
where ( , , )x t k ∈Ω× × ℑI and for the sake of simplicity the source term is assumed constant, i.e. f =cte. Because the conductivity is considered unknown, it is assumed as a new coordinate defined in the interval ℑ. Thus, instead of solving the thermal model for different values of the conductivity parameter we prefer introducing it as a new coordinate. The price to be paid is the increase of the model dimensionality; however, as the complexity of PGD scales linearly with the space dimension the consideration of the conductivity as a new coordinate allows for faster and cheaper solutions.
The solution of Eq. (8) is searched under the form:
(
)
( ) ( ) ( )
1 , , i N i i i i u t k X T t K k = = ≈∑
⋅ ⋅ x x (9)In what follows we are assuming that the approximation at iteration n is already done:
(
)
( ) ( ) ( )
1 , , i n n i i i i u t k X T t K k = = =∑
⋅ ⋅ x x (10)and at present iteration we look for the next functional product Xn+1
( )
x ⋅Tn+1( )
t ⋅Kn+1( )
k that foralleviating the notation will be denoted by R
( ) ( ) ( )
x ⋅S t W k⋅ . Prior to solve the resulting non linear model related to the calculation of these three functions a model linearization is compulsory. The simplest choice consists in using an alternating directions fixed point algorithm. It proceeds by assuming S t( )
and W k( )
given at the previous iteration of the non-linear solver and then computing R x( )
. From the just updated R x( )
and W k( )
we can update( )
S t , and finally from the just computed R x
( )
and S t( )
we compute W k( )
. The procedure continues until reaching convergence. The converged functions R x( )
, S t( )
and W k( )
allow defining the searched functions: Xn+1( )
x =R( )
x , Tn+1( ) ( )
t =S t and Kn+1( )
k =W k( )
.We are illustrating each one of the just referred steps:
I. Computing R x from
( )
S t and( )
W k :( )
We consider the global weak form of Eq. (8):* 0 I u u k u f d dt dk t Ω× ×ℑ ∂ − ∆ − = ∂
∫
x (11)where the trial and test functions write respectively:
(
)
( ) ( ) ( ) ( ) ( ) ( )
1 , , i n i i i i u t k X T t K k R S t W k = = =∑
⋅ ⋅ + ⋅ ⋅ x x x (12) and(
)
( ) ( ) ( )
* * , , u x t k =R x ⋅S t W k⋅ (13)Introducing (12) and (13) into (11) it results
* * 1 1 I i n i n i i i i i i i i I S R S W R W k R S W d dt dk t T R S W X K k X T K f d dt dk t Ω× ×ℑ = = = = Ω× ×ℑ ∂ ⋅ ⋅ ⋅ ⋅ ∂ ⋅ − ⋅∆ ⋅ ⋅ = ∂ = − ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅∆ ⋅ ⋅ − ∂
∫
∑
∑
∫
x x (14)Now, being known all the functions involving the time and the parametric coordinate, we can integrate Eq. (14) in their respective domains I× ℑ. Integrating in I× ℑ and taking into account the notation
2 2 2 1 1 1 2 2 2 2 3 3 3 4 4 4 5 5 5 I I I i i i i i i I i i i i i i I w W dk s S dt r R d dS w kW dk s S dt r R R d dt w W dk s S dt r R d dT w W K dk s S dt r R X d dt w kW K dk s S T dt r R X d ℑ Ω ℑ Ω ℑ Ω ℑ Ω ℑ Ω = = = = = ⋅ = ⋅∆ = = = = ⋅ = ⋅ = ⋅∆ = ⋅ = ⋅ = ⋅
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
x x x x x (15)Eq. (14) reduces to:
(
)
* 1 2 2 1 * 4 4 5 5 3 3 1 1 i n i n i i i i i i i i R w s R w s R d R w s X w s X w s f d Ω = = = = Ω ⋅ ⋅ ⋅ − ⋅ ⋅ ∆ = = − ⋅ ⋅ ⋅ − ⋅ ⋅ ∆ − ⋅ ⋅ ∫
∑
∑
∫
x x (16)Eq. (16) defines an elliptic steady state boundary value problem that can be solved by using any discretization technique operating on the model weak form (finite elements, finite volumes …). Another possibility consists in coming back to the strong form of Eq. (16):
1 2 2 1 4 4 5 5 3 3 1 1 i n i n i i i i i i i i w s R w s R w s X w s X w s f = = = = ⋅ ⋅ − ⋅ ⋅∆ = − ⋅ ⋅ − ⋅ ⋅∆ − ⋅ ⋅
∑
∑
(17)that could be solved by using any collocation technique (finite differences, SPH …).
II. Computing S t from
( )
R x and( )
W k :( )
In the present case the test function writes:(
)
( ) ( ) ( )
* *
, ,
u x t k =S t ⋅R x ⋅W k (18)
Now, the weak form reads
* * 1 1 I i n i n i i i i i i i i I S S R W R W k R S W d dt dk t T S R W X K k X T K f d dt dk t Ω× ×ℑ = = = = Ω× ×ℑ ∂ ⋅ ⋅ ⋅ ⋅ ∂ ⋅ − ⋅∆ ⋅ ⋅ = ∂ = − ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅∆ ⋅ ⋅ − ∂
∫
∑
∑
∫
x x (19)that integrating in the space Ω× ℑ and taking into account the notation (15) results:
* 1 1 2 2 * 4 5 5 4 3 3 1 1 I i n i n i i i i i i i i I dS S w r w r S dt dt dT S w r w r T w r f dt dt = = = = ⋅ ⋅ ⋅ − ⋅ ⋅ = = − ⋅ ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅
∫
∑
∑
∫
(20)Eq. (20) represents the weak form of the ODE defining the time evolution of the field S that can be solved by using any stabilized discretization technique (SU, Discontinuous Galerkin, …). The strong form of Eq. (20) reads:
1 1 2 2 4 5 5 4 3 3 1 1 i n i n i i i i i i i i dT dS w r w r S w r w r T w r f dt dt = = = = ⋅ ⋅ − ⋅ ⋅ = − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅
∑
∑
(21)than can be solved by using backward finite differences, or higher order Runge-Kutta schemes, among many other possibilities.
III. Computing W k from
( )
R x and( )
S t :( )
In the present case the test function writes:
(
)
( ) ( ) ( )
* *
, ,
u x t k =W t ⋅R x ⋅S k (22)
Now, the weak form reads * * 1 1 I i n i n i i i i i i i i I S W R S R W k R S W d dt dk t T W R S X K k X T K f d dt dk t Ω× ×ℑ = = = = Ω× ×ℑ ∂ ⋅ ⋅ ⋅ ⋅ ∂ ⋅ − ⋅∆ ⋅ ⋅ = ∂ = − ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅∆ ⋅ ⋅ − ∂
∫
∑
∑
∫
x x (23)that integrating in the space Ω×I and taking into account the notation (15) results:
(
)
* 1 2 2 1 * 5 4 4 5 3 3 1 1 i n i n i i i i i i i i W r s W r s W dk W r s K r s K r s f dk ℑ = = = = ℑ ⋅ ⋅ ⋅ − ⋅ ⋅ = = − ⋅ ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ ∫
∑
∑
∫
(24)Eq. (24) does not involve any differential operator. The strong form of Eq. (24) reads:
(
1 2 2 1)
(
5 4 4 5)
3 3 1 i n i i i i i i r s r s W r s r s K r s f = = ⋅ − ⋅ ⋅ = − ⋅ − ⋅ ⋅ − ⋅ ⋅ ∑
(25)that represents an algebraic equation. Thus, the introduction of parameters as additional model coordinates has not a noticeable effect in the computational cost.
There are other minimization strategies more robust and exhibiting faster convergence [10] for building-up the PGD.
2. APPLICATIONS
In this section we are analyzing different applications of the proper generalized decomposition in computational science and engineering, and in particular in material and processes.
2.1 Materials description at their finer scales: quantum chemistry
In quantum mechanics one is confronted with the solution of Schrödinger equation giving the electronic and nuclei distribution. If we assume a system composed of Np particles (electrons
and nuclei), the evolution of the joint wavefunction Ψ = Ψ
(
x x1, 2, ,xNp,t)
⋯ is governed by the
Schrödinger equation whose dimensionless form in absence of relativistic and spin effects, writes: 2 1 2 1 1 p p p p N p N k N p pk p p p k k p i V t m = = = = = = > ∇ Ψ ∂Ψ = − + Ψ ∂
∑
∑ ∑
(26)where each particle is defined in the whole physical space 3
j∈
x R ,i= −1 and ℏ represents the
Planck’s constant divided by 2π . The differential operator 2
p
∇ is defined in the conformation space of each particle, i.e.: 2 2 2 2 2 2 2
p xp yp zp
∇ = ∂ ∂ +∂ ∂ +∂ ∂ . The Coulomb’s potential accounting for the inter-particles interactions writes:
p k pk p k q q V = × − x x (27)
where the masses mp are unity for electrons, the charges qj are −1 for electrons and +Zj (atomic
numbers) for nuclei.
The wavefunction function is then defined in a highly multidimensional space:
(
)
3 1, 2, , , : p p N N t × +Ψ = Ψ x x ⋯ x ℝ ×ℝ →ℂ. Due to the curse of dimensionality, illustrated above, its
direct solution has been only possible for very reduced quantum systems composed of some particles (one or two nuclei and very few electrons). The PGD lies in searching a solution representation in the form:
(
1 2)
1 ( )1( )
( ) 1 , , , , p p p i N i i N N N i i t F F T t = = Ψ = Ψ x x ⋯ x ≈∑
x ⋯ x ⋅ (28)In [6] we describe deeply the solution of quantum chemistry models by using the PGD.
2.2 Materials description at their finer scales: kinetic theory descriptions
In the kinetic theory framework the equation governing the evolution of conformation
distribution is known as the Fokker-Planck equation. This equation results from a simple conservation balance of such distribution function that is defined in the physical space ( )x, t and uses some others coordinates qi –conformational coordinates - for describing the internal
structure (e.g. the vectors defining the different arms of molecules ….). The balance equation writes:
( )
(
( )
)
1 j N j j t = ∂Ψ + ∇ ⋅ Ψ = − ∇ ⋅ Ψ ∂ x v∑
q qɺ (29)that usually results defined in highly multidimensional spaces. A way for circumventing the curse of dimensionality consists in applying the PGD:
( 2 ) ( ) ( )1 1 ( ) ( ) 1 , , , , i N i i N i N N i i t X Q Q T t = = Ψ = Ψ x q ⋯q ≈
∑
x ⋅ q ⋯ q ⋅ (30)as was successfully performed in some of our former works [1][2][3][9].
2.3 Parametric models
We consider the 1D heat equation defined by:
( , ); t, x u u k f t x t x t x x ∂ − ∂ ∂ = ∈Ω ∈Ω ∂ ∂ ∂ (31)
If k is constant, this equation can be solved for every value of k using the separated representation previously illustrated without further difficulties. Now, we are focusing on a more complicated and realistic problem. In general, for homogenous materials the thermal conductivity depends on temperature, following a linear dependence:
k =au+b (32)
If we introduce this expression into Eq. (31) the resulting heat equation writes: 2 2 2 2 2 u u u u b au a f t x x x ∂ ∂ ∂ ∂ − − − = ∂ ∂ ∂ ∂ (33)
We want to solve this equation for any value of a and b. We can easily understand that the computed solution allows efficient optimization and inverse identification strategies. For that purpose the following separated representation approximation is considered:
1 ( , , , ) ( ) ( ) ( ) ( ) N i i i i i u t x a b T t X x A a B b = ≈
∑
⋅ ⋅ ⋅ (34)The numerical solution was carried out for t∈
[ ]
0,1 , x∈ −[ ]
1,1 , a∈ −[ ]
1,1 , b∈[ ]
1,5 being the initial condition u t( =0, ) 1x = −x10, the temperature vanishing at both boundaries. 500 nodes were employed for discretizing each of the 4 coordinates. A mesh would have involved 5004 degrees of freedom, but using the PGD this impressive number reduces to the order of 2000! The solution was performed in few minutes using Matlab on a personal laptop. Fig. 1 depicts the solution at point x=0 at the final time t=1 as a function of both parameters a and b.The non linearity was accounted by assuming the conductivity given at previous iteration, i.e. n k =au +b, where 1 ( , , , ) ( ) ( ) ( ) ( ) n n i i i i i u t x a b T t X x A a B b = =
∑
⋅ ⋅ ⋅ (35)Other linearizations were analyzed and discussed in [3] and [10].
Fig. 1- Temperature at point x=0 at the final time t=1 as a function of both parameters a and b.
Now, we come back to the non-linear thermal model (31), with a=1 and b=1, but we are focusing on its steady solution for a source term given by f = −β
(
1−x2)
, β∈[ ]
0,1 . Now, the solution is searched from the finite sum decomposition:1 ( , ) ( ) ( ) N i i i u x β X x B β = ≈
∑
⋅ (36)The computed solution is depicted in figure 2 where we can notice, as expected, that the solution vanishes everywhere when β =0.
0 0.5 1 -1 -0.5 0 0.5 1 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1
Fig. 2- Steady state temperature field u x( , )β , x∈ −
[ ]
1,1 and β∈[ ]
0,1 .Finally, we are considering the same problem, the steady state solution of a thermal model (non-linear) in which the heat source is punctual and can be applied everywhere in the domain. The thermal model is then defined as follows:
( )
u k f x x x ∂ ∂ = ∂ ∂ (37)where the source term writes in the present case:
(
)
( ) '
f x = ⋅β δ x−x (38)
being ()δ the Dirac’s mass, 'x the point in which the thermal load of intensity β applies. As we are interested in solving the model (37) for any position of the thermal load 'x and any value of the intensity β , we introduce the load location and the load intensity as new coordinates, searching the solution in the form:
( )
1 ( , ', ) ( ) ' ( ) N i i i i u x x β X x S x B β = ≈∑
⋅ ⋅ (39)By applying the procedure described in the previous section one could determine easily the functions X xi( ) by solving a second order elliptic problem, then function S xi( ') by solving an
algebraic equation (there are not derivatives in that coordinate in the departing model) and finally function Bi( )β solving another algebraic equation.
The strategy here described allows an off-line pre-calculus and very fast post-processing required in many branches of computational sciences with real-time simulation purposes.
2.4 Local models
Many models involve local kinetic models. Thus, for example in presence of reacting substances, the concentration of each species is governed by a diffusion equation that contains a source term taking into account of the chemical reactions. In some cases the number of reactions involved could be important. Let’s R be the number of the reactions, and M the number of nodes involved in the domain discretization. Thus, if one makes use of standard strategies, the R kinetic reactions must be solved at the M nodes. Despite the simplicity that the integration of local models implies, it can be expensive in computational resources. In what follows we are describing some alternatives for alleviating the just referred computational issue.
( )
1 , , 1, , i R i ij j i dC x C i i R dt α = = =∑
⋅ ∀ = ⋯ (40)It is clear that at each node x , the kinetic coefficients k αij
( )
xk become well defined, and then Eq. (40) can be easily integrated by applying for example a backward finite difference scheme. Thus one should perform R M× 1D time integrations.A first alternative for reducing the computing cost lies in a globalization of the local problem that consists in enforcing the decomposition:
( )
1 ( , ) ( ) N i i i i C x t X x T t = ≈∑
⋅ (41)that introduced into (40) leads to the solution of R 2D problems.
We proposed recently [7] the use of a more efficient decomposition that writes:
( ) ( )
1 ( , , ) ( ) N i i i i C x t c X x T t C c = ≈∑
⋅ ⋅ (42)where the third dimension has a discrete nature, taking integer values in the interval
[
1, ,]
c∈ ⋯ R ⊂ℕ . Now, equation (40) can be easily rewritten to include C x t c as main ( , , )
unknown field and solved leading to the decomposition (42). Thus, by using the present formulation only one 3D problem must be solved for computing the evolution of all the species concentrations at all locations.
2.5 Evolving domains: toward efficient simulations of material forming processes
The main issue in treating models defined in evolving domains lies in the fact that x belongs
to a domain that is evolving in time, i.e. x∈Ω
( )
t . Obviously, in that case we cannot apply the procedure described previously, but the most deep difficulty lies in the fact that the space coordinate is not independent of the time coordinate.The simplest alternative for circumventing this difficulty consists of defining the map between the initial space coordinates X∈Ω =
(
t 0)
and the present ones x∈Ω( )
t . This mapping writes: x=x X( )
,t . Now, the model given by Lx t,(
u( )
x,t)
=F (where Lx t,( )
denotes a differential operator involving the present coordinates x and t ) is redefined by considering( )
X,t as new coordinates (now both being independent): L*X,t(
u( )
X,t)
=F and solved by using * the natural decomposition:( )
1 ( , ) ( ) N i i i u t X T t = ≈∑
⋅ X X (43)We are at present analyzing this approach that suggests fully Lagrangian formulation in thermo-mechanical problems. As the procedure described in section 1.2 do not enforce the balance equations at any particular time, fully Lagrangian formulations could run as soon as the mapping x=x X
( )
,t is well defined even in very large geometrical transformations.2.6 Shape optimization
Closely related to the discussion carried out just before, in this section we are considering another exciting topic, the one related to the shape optimization. In that case the domain geometry depends on a set of parameters
(
p1,⋯,pP)
that should be chosen in order to reach theoptimum of a given cost function. Obviously, as Ω depends on the parameters, i.e.
(
p1, ,pP)
Ω ⋯ the space and the parametric coordinates are no more independent. For this
reason, we consider the reference configuration X∈Ω*
(
p1 =0,⋯,pP =0)
and define the mapping: x=x X(
,p1,⋯,pP)
. Now, the model is rewritten considering the space and parametric coordinates, looking for a solution in the separated form:( )
( )
1 1 1 1 ( , , , ) ( ) N i i P i P P i u p p X P p P p = ≈∑
⋅ X ⋯ X ⋯ (44)that allows solving the model for all possible values of the shape parameters. Now, the cost function can be evaluated and the optimal value identified. We are analyzing the optimality conditions in order to compute the global maximum (or minimum) of the cost function instead the usual relative ones.
2.7 Multi-scale models
Consider the simple model
( )
( ) ( )
] ]
2 2
2 cos 2 cos sin , 0,10
du
t t t t t t I
dt = ω − ω ω ω ∈ = (45)
with zero initial condition and whose exact solution writes uex =t2cos2
( )
ωt . When the frequency increases, the time step must be reduced in order to capture the load evolution. Fig. 3 compares for a frequency ω=10the computed solution by using a time step of ∆ =t 0.001(high resolution curve) and ∆ =t 0.5(low resolution curve).0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100
Fig. 3- Solution of model (45) with two different sampling times.
For accounting with the two time scales involved in the model we introduce two different times, both assumed independent: T = ∈t I and τ ω= t∈
[
0, 2π]
. Thus, we postulate u T( )
,τ and Eq. (45) becomes:( )
, 2( )
2( ) ( )
2 cos 2 cos sin
du T u T u u u T T dt T t t T τ τ ω τ ω τ τ τ τ ∂ ∂ ∂ ∂ ∂ ∂ = ⋅ + ⋅ = + = − ∂ ∂ ∂ ∂ ∂ ∂ (46)
whose solution is searched by assuming the decomposition:
( )
1 ( , ) ( ) N i i i u T τ F T G τ = ≈∑
⋅ . Obviously, in more complex multi-scale models involving different scales in space ( ,x x ⋯ 1 2, ) and time ( , ,t t ⋯ we should write: 1 2 ) u t t( , ,1 2 ⋯,x x1, 2,⋯ , and in that case a separated )representation seems compulsory. In the present case the decomposition involves a single product of functions u T( , )τ =F T G( )⋅
( )
τ =T2⋅cos2( )
τ that correspond with the exact solution, both depicted in Fig. 4. In that figure we depict also the reconstructed solutionu T( , )τ .0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
Fig. 4- Functions F T (left) and ( )( ) Gτ (centre) associated with the decomposition of u T( , )τ in (46) and well as the reconstructed solution u T( , )τ (right).
3. REFERENCES
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[2] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations. J. Non-Newtonian Fluid Mech., 144 (2007), 98-121.
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