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Multiresolution Analysis in Radiation Transport

HyeongKae Park

and

Cassiano R. E. de Oliveira

Nuclear and Radiological Engineering Program,

George W. Woodruff School of Mechanical Engineering,

Georgia Institute of Technology

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Outline

- Introduction/Motivation
- FE-PN equations
- Spatial adaptivity
- Angular adaptivity
- Numerical examples

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Introduction/Motivation

- Importance of numerical simulation as a design tool.
- Use of deterministic computational simulation to predict the complex radiation fields is becoming increasingly prevalent (new generation of nuclear reactors, space reactors, and multi-physics applications).
- Attempts to simulate ever-larger, more complex problems require efficient and robust numerical algorithmic development.

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Introduction/Motivation

- Accurate and efficient numerical methods for modeling (large) complex systems.
- Accuracy of the numerical solution depends on the discretization methods of the independent variables (space, angle, and energy).
- The necessary space-angle resolution required to achieve a desired local accuracy varies throughout the problem domain.

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Introduction/Motivation

- Q: What is the resolution that I should use to solve my radiation transport problem numerically
- A: I don’t know – let the method decide

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Introduction/Motivation

- Adaptivity as the way forward
- Usually, there is no prior knowledge about the resolution requirement to meet the prescribed accuracy. Especially if the problem itself is not known ie inverse problem
- Spatial and angular variable are tightly coupled in radiation transport problems. Thus, space-angle coupled adaptivity must be considered (not covered in this talk).

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Classical Finite Element-Spherical Harmonics (FE-PN) Discretization Method

1st order radiation transport equation (steady state, one speed):

2nd order even-parity form of the radiation transport equation:

where,

Even-parity flux

Parity source

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

FE-PN Equations

- C and G is the integral form of collision operators:
- The second order formulation is self-adjoint. Thus we can use an extremum variational principle to formulate FE-PN equations (Ritz-Galerkin method).

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

FE-PN Equations

- Weak form of the parity equation:
- Angular flux approximated by the trial function of the form:
- Discretized form of functional:

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

FE-PN Equations

- Ritz-Galerkin procedure yields system of equations of the form:
- The matrix A has a shell structure (for =0, 2, 4,…):

“Diffusion” matrix stencil

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Adaptive Strategy

- Discretization of system of equations
- Solution of discretized system of equations
- Measure of the discretization error
- Refinement/creation of new discrete system

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Posteriori Error Measurement

- Posteriori error measurement provides information where the discretization needs to be altered (refinement or coarsening) to achieve desired accuracy.
- Two basic strategy to compute posteriori error measure.
- Derivative recovery error estimator (gradient 1st or Hessian 2nd)
- Residual based error estimator

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Length of the element edge respect to metric M

Shape of element

Spatial Adaptivity- Popular choice of error estimator is the derivative recovery.
- Gauge of the mesh quality is measured by the length of the element edge and shape of the element.

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Hessian Based Spatial Adaptivity

- Hessian based adaptivity is suitable for the anisotropic adaptivity
- Expanding the solution in Taylor series:
- The leading term of the interpolation error ( for linear finite elements)
- Interpolation error is defined as:

(H=Hessian Matrix)

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Hessian Based Spatial Adaptivity

- Hessian Matrix H is outer product of gradient operator.

Hessian matrix provides the idea of directional dependent error measure.

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Hessian Based Spatial Adaptivity

- Hessian matrix can be decomposed into:
- Direction of eigenvector with large eigenvalue (magnitude) corresponds to the direction associated with large error.

Eigenvector

Eigenvalue

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Approximating Hessian Matrix

- Galerekin methods are repeatedly applied to calculate first derivative at node.
- The Hessian term can be calculated applying Galerkin methods to the calculated first derivative.
- L is row summed lumped mass matrix
- N is the finite element basis function

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Angular Adaptivity

- The degree of anisotropy in angular flux varies widely with space and energy groups.
- Highly-anisotropic angular flux region may be largely localized.
- Uniform refinement is not computationally efficient.
- Local refinement of angular expansion is more desirable.

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Angular Adaptivity

- Variable angular resolution can be achieved by truncating the angular expansions nodal-wise.
- The hierarchical nature of spherical harmonics expansion allows addition of higher moments shell without changing the existing matrix.
- Error can be estimated by computing residual of new system with old solution.

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Error Measurement for Angular Expansion

- Main quantity of interest in radiation transport problem is the scalar flux:
- The error and change in error in scalar flux is:
- Thus, it is desirable to estimate the change in scalar flux due to the addition of the higher spherical harmonics moment. If the change is small the contribution of the higher moment is negligible. Therefore, we can truncate the angular expansion in that region.

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Error Estimate Using Residual

- Scalar flux can be found by:
- Then change in error from PN to PN+1 can be approximated by:
- The residual of the system going from PN to PN+1 is:

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Algorithm

- For L = 1, N
- Solve PL equation
- Temporary add the PL+1 shell
- Calculate residual
- For n=1,NODES
- Scale residual with diagonal coefficient
- If magnitude of scaled residual is small enough discard the nodes
- Otherwise keep the node.
- Go to next level

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Numerical Example (Advanced Gas-Cooled Reactor)

- Spatial adaptivity using Hessian error measure

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Numerical Example ( photon propagation)

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Numerical Example (nuclear reactor lattice)

- 2D C5G7MOX benchmark problem
- 4 fuel assemblies
- Water moderator
- 7 groups
- Criticality problem

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Numerical Example (P15 approximation)

Group 1

Group 4

Group 7

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Numerical Example (% error)

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Numerical Example

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

Conclusions

- Adaptive strategies are needed for efficient numerical simulations.
- Derivative-based error measure for spatial adaptivity, and residual-based error measure for the angular adaptivity are suitable for the FE-PN equations.
- Efficient reduction in degrees-of-freedom is possible using point-wise adaptive spherical harmonics expansion.
- Challenge is fully coupled space-angle adaptivity

19th International Conference on Transport Theory, Budapest, Hungary, July 24-30 2005

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