OUTLINES

A component mode synthesis method for 3D cell by cell calculation using the mixed dual finite element solver MINOS

P. Gurin, A.M. Baudron, J.J. LautardCommissariat lEnergie AtomiqueDEN/DM2S/SERMACEA SACLAY91191 Gif sur Yvette Cedex [email protected]

American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005

OUTLINESGeneral considerations and motivationsBasic equationsMINOS SolverThe component mode synthesis methodNumerical resultsConclusions and perspectives


General considerations and motivations Basic equations MINOS Solver Numerical results Conclusions and perspectives The component mode synthesis method


Geometry and mesh of a PWR 900 MWe core


INTRODUCTIONMINOS solver :main core solver of the DESCARTES system, developed by CEA, EDF and Framatomemixed dual finite element method for the resolution of the SPn equations in 3D cartesian homogenized geometries3D cell by cell homogenized calculations too expensive

Standard reconstruction techniques to obtain the local pin power can be improved for MOX reloaded coresinterface between UOX and MOX assemblies


MOTIVATIONSFind a numerical method that takes in account the heterogeneity of the core

Domain decomposition and two scale method : Core decomposed in multiple subdomainsProblem solved with a fine mesh on each subdomainGlobal calculation done with a basis that takes in account the local fine mesh results

Perform calculations on parallel computers




Strong formulation of SPN equations Derived from 1D transport Pn equationN+1 harmonics : The (N+1)/2 even components are scalar The (N+1)/2 odds components are vectors

SPN one group equation written in the mixed form (odd even) with albedo boundary condition reads :Coefficients :


Mixed dual variational SPN formulationBy projection and using the Green formula on the odd equations :Even flux : discontinuousOdd flux : normal trace continuous


Existence and unicity of the solutionMixed dual variational SPN equations are a particular case of the more abstract problem :

The ellipticity of the bilinear continuous form a and the inf-sup condition on the continuous form b insure existence and unicity of the solution of this problem :




Discretized spacesRTk basis with :

Even basis => Orthogonal lagrangian basis associated to nodes located at Gauss points of order 2k+1 Odd flux basis such that :Finite Element basis on rectangle : Raviart Thomas Nedelec element (RTk)


The matrix systemThe matrix of the discretized system is :Block Gauss Seidel iteration (1 block corresponds to the set of nodes of one odd flux component)Eigenvalue problem solved by power iterations




The CMS method CMS method for the computation of the eigenmodes of partial differential equations has been used for a long time in structural analysis.

The steps of our method : Decomposition of the core in K small domainsCalculation with the MINOS solver of the first eigenfunctions of the local problem on each subdomain All these local eigenfunctions span a discrete space used for the global solve by a Galerkin technique


Diffusion modelMonocinetic diffusion problem with homogeneous Dirichlet boundary condition.

Mixed dual weak formulation : Eigenvalue problem


Local eigenmodesOverlapping domain decomposition :

Computation on each of the first local eigenmodes with the global boundary condition on , and p=0 on \ :


Global Galerkin methodExtension on E by 0 of the local eigenmodes on each : global functional spaces on EGlobal eigenvalue problem on these spaces :


Linear systemUnknowns :with :Linear system associated :


Global problemGlobal problem :

H symmetric but not positive definite

Not always well posed because of the inf-sup condition increase the number of odd modes




Domain decompositionDomain decomposition in 201 subdomains for a PWR 900 MWe loaded with UOX and MOX assemblies :Internal subdomains boundaries :on the middle of the assembliescondition p=0 is close to the real valueInterface problem between UOX and MOX is avoided


Power and scalar flux representationdiffusion calculationtwo energy groupscell by cell meshRTo element


Comparison between our method and MINOS : 2DKeff difference, and norm of the power difference between CMS method and MINOS solutionMore odd modes than even modes inf-sup conditionTwo CMS method cases : 4 even and 6 odd modes on each subdomain9 even and 11 odd modes on each subdomain

4 modes9 modes keff (pcm)4.41.4

5 %0.92 %


Comparison between our method and MINOS : 2DPower gap between CMS method and MINOS in the two cases. Normalization factor : 4 even modes, 6 odd modes9 even modes, 11 odd modes


Comparison between our method and MINOS : 2DPower cell difference between CMS method and MINOS solution in the two cases. Total number of cells : 334084.4 even modes, 6 odd modes95% of the cells : power gap < 1%

9 even modes, 11 odd modes95% of the cells : power gap < 0,1%

Graph1

5294

13406

28576

226087

38890

12052

5118

Power difference

Number of cells

Feuil1

-0.0115294

-0.007713406

-0.00428576

-0.0003226087

0.003338890

0.00712052

0.0115118

Graph1

3217

10511

34965

244898

33210

4591

814

Power difference

Number of cells

Feuil1

-0.00193217

-0.001210511

-0.0005534965

0.0001244898

0.0007533210

0.00144591

0.0021814


3D resultsSame domain decomposition than in 2D.Keff difference, and norm of the power difference between CMS method and MINOS solution :The core is split into 20 planes in the Z-axis :Two CMS method cases : 4 even and 6 odd modes on each subdomain8 even and 10 odd modes on each subdomain

4 modes8 modes keff (pcm)7.32.5 5.1 %1 %


Comparison between our method and MINOS : 3DPower cell difference between CMS method and MINOS solution in the two cases. Total number of cells : 6681680.4 even modes, 6 odd modes95% of the cells : power gap < 1%8 even modes, 10 odd modes90% of the cells : power gap < 0,1%

Graph1

99362

265782

617300

4801175

586180

176047

70978

Power difference

Number of cells

Feuil1

-0.01299362

-0.0082265782

-0.0043617300

-0.000344801175

0.0036586180

0.0075176047

0.01170978

Graph1

56406

162732

413833

4638241

968296

286730

71875

Power difference

Number of cells

Feuil1

-0.00356406

-0.002162732

-0.0011413833

-0.000134638241

0.00081968296

0.0017286730

0.002771875


CPU time and parallelizationSo far MINOS solver is faster than CMS method, BUT :The code is not optimizedThe deflation method used by the local eigenmodes calculations in MINOS can be improved

CMS method most of the time spent in local calculationsIndependent calculations, need no communication on parallel computersMatrix calculations are easy to parallelize too.Global solve time is very smallWith N processors, we expect to divide the time by almost N On parallel computer, the CMS method will be faster than a direct heterogeneous calculation




Conclusions and perspectivesModal synthesis method :Good accuracy for the keff and the local cell powerWell fitted for parallel calculation : the local calculations are independent they need no communication

Future developments : Parallelization of the codeExtension to 3D cell by cell SPn calculationsPin by pin calculationComplete transport calculations

This presentation concerns a new technique for reconstruction of the pin power for heterogeneous 3D SPn calculation.This work is made in the framework of my PhD. My director are Jean-Jacques Lautard in CEA Saclay and Yvon Maday in Paris VI University.The method is based on domain decomposition with overlapping subdomains and a modal synthesis technique for the global flux reconstruction.It uses the solver MINOSAfter general considerations and motivations of our work,

I will recall briefly simplified transport equation , and I will speak about the MINOS solver who uses mixed dual equation.

I will explain our component mode synthesis method and I will present the first numerical results in 2D and 3D.

Finally I will conclude and present our future goals

After general considerations and motivations of our work,




We can see on this pictures the three scales of a core : pin, assembly and the whole core.

The mesh of the core must be very fine in order to take in account all the heterogeneities of the core.

Our motivations are :Find a numerical method who take in account the heterogeneity of the coreUse a domain decompositionSolve the problem in three steps and proceed as a two scale method :Decompose the core in multiple subdomainSolve the problem with a fine mesh on each subdomainMake the global calculation with a functional basis who take in account the local fine results.Implement the code on parallel computer

Our motivations are :Find a numerical method who take in account the heterogeneity of the coreUse a domain decompositionSolve the problem in three steps and proceed as a two scale method :Decompose the core in multiple subdomainSolve the problem with a fine mesh on each subdomainMake the global calculation with a functional basis who take in account the local fine results.Implement the code on parallel computer





I recall here the strong formulation of the simplified Pn equations, derived from 1D transport Pn equation.

The angular variable of the transport equation is descretized on a polynomial Legendre basis.

N+1 harmonics : The even components are scalarThe odd components are vectors

The SPn one group equation written in the mixed form (odd-even components) with albedo boundary condition reads :

Where H is a tridiagonal matrix coupling the harmonics, a full matrix which depends on the albedo coefficients, Te and To respectively the even and odd removal diagonal matrices.One obtain the weak mixed dual formulation of the SPn equation from the strong formulation, by projection on the odd space and the even space and using the Green formula on the odd equation :

Also the even flux is discontinuous and the normal trace of the odd flux is continuous.

Mixed dual variational SPN equations are particular case of the more abstract problem :

The ellipticity of the bilinear continuous form a and inf-sup condition on the continuous form b insure existence and unicity of the solution of this problem :





The geometry of the core is rectangular, and the Raviart-Thomas-Nedelec element is choosen for the finite element basis.

The even basis is orthogonal, using Lagrangian polynomials. The associated nodes are the Gauss points.

The odd flux basis is choosen in order to have a difference matrix B :

Elementary matrices are estimated by numerical integration at the even flux nodes : it simplifies the calculation and produces superconvergence

The global matrix of the discretized system is :

This matrix is not symmetric positive definite, but after the elimination of the even flux, the linear system on the odd flux to solve is :

The solver perform a block Gauss-Seidel iteration : one block is composed by the set of nodes of one component of the odd fluxes

The case of the eigenvalue problem is solved by the external power iterations technique





The CMS method can be decomposed in 3 steps :

First the decomposition of the core in K small domains.

Second the calculation with the MINOS solver of the first eigenfunctions of the local problem (not only the first one) on each subdomain. We use reflective boundary conditions on internal interfaces, the actual core boundary conditions are used on the external boundary.

Third the set of this local spatial eigenfunctions is extended to the global domain by supposing zero flux outside the subdomain. These functions are used to span a discrete space that allows fundamental mode approximation through a mixed dual Galerkin technique.

For the sake of simplicity, we will explain in detail the monocinetic diffusion case with homogeneous Dirichlet boundary condition .The approach is the same in the SPn multigroup general case, even with other boundary conditions.

The mixed dual weak formulation of this problem reads :The first step of our CMS method is the local solve.

The core is decomposed in overlapping subdomain :

On each subdomain the first eigenmodes are computed by MINOS with the global boundary condition on the external boundaries and infinite medium boundary conditions on the internal one :All this local eigenmodes are used in order to solve the global problem on the wall core with a Galerkin technique :

The local eigenmodes on each subdomain are extended on the global domain by 0 in order to have global functions defined on the wall core.

So we obtain approximated spaces :the odd space is included in Hdiv, thanks to the reflective condition on internal boudaries of the subdomain,the even one is included in L2.

Projection of the global problem on this spaces reads:

The unknowns associated to the new global functional basis can be written :

And the discretized problem reads :

The linear system associated reads :Finally the global system we have to solve is :

As in the MINOS solver, the global matrix H is symmetric but not positive definite. In our case we dont eliminate the even unknowns, because the matrix Ta is not diagonal.

Unfortunately this problem is not well posed because the inf-sup condition is not always verified. One technique to enforce this condition is to increase the number of odd modes with respect to the even ones.





Fig. 1b and c represents the proposed decomposition in 201 subdomains for a PWR 900 MWe core calculation (Fig. 1a).

We have chosen the internal subdomains boundaries on the middle of the cells, where the condition p=0 is close to the real value. Furthermore with this decomposition we avoid the interface problem between MOX and UOX assemblies, because the interface is in the subdomain, not on the boundary.

Each subdomain resolution is independent, so the future CMS implementation in parallel will be simple and very efficient : there is no communication between the local solvers.A computer code performs the CMS method for 3D core calculations; it is based on the existing MINOS solver. In order to validate the method, we present here results for 2D diffusion calculation with two energy groups (Fig.).

Table 1 present Keff difference and L2 and Linf norm of the power difference between the whole core calculation by MINOS and our CMS method (with the decomposition presented on Fig. 1b and 1c) in two cases: 4 even and 6 odd modes on each subdomain in the first case, 9 even and 11 odd modes in the second case.

The number of the odd modes must be larger than the number of even modes, probably for the reason mentioned above (inf-sup condition verified in this case).

Here is the representation of the power difference on the core between the MINOS calculation and our CMS method in the two cases.

We can see the error at the interface between the subdomains.

Now the histogram for the power cell difference between CMS method and MINOS solution in the two cases.

The total number of cells is 334084.

In the first case, for 95% of the cells, the power gap is less than 1%.In the second case, for 95% of the cells, the power gap is less than 0.1%.

I present here results in 3D.

The reactor core is the equivalent of the 2D one. It is made out of 20 plans. The first one and the last one is reflector, the other are made with the same assemblies than in the 2D core.

The decomposition in 201 subdomains is the same than the 2D one.

Table 2 presents Keff difference and L2 and Linf norm of the power difference between MINOS and our CMS method in two cases: 4 even and 6 odd modes on each subdomain in the first case, 8 even and 10 odd modes in the second case.

Now the histogram for the power cell difference between CMS method and MINOS solution in the two cases.

The total number of cells is now 6681680.

In the first case, for 95% of the cells, the power gap is less than 1%.In the second case, for 90% of the cells, the power gap is less than 0.1%.

In conclusion, the modal synthesis has several favors :

As shown, we obtain a good convergence and accurate calculation even with only a few modes. The precision for the keff is very good (

OUTLINES

Documents

Transcript of OUTLINES