Development of Multi-level Reduced Order MOdeling Methodology
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This document summarizes the development of a new multi-level reduced order modeling (MLROM) methodology. MLROM allows extracting the effective dimensionality of a high-fidelity nuclear reactor model by executing the model in a small sub-domain, like a pin cell, rather than the entire domain. This significantly reduces computational cost compared to previous ROM methods. The document describes the mathematical framework of MLROM and presents initial numerical tests of the method using a benchmark boiling water reactor lattice model. Results suggest MLROM can accurately represent the full model while requiring far fewer executions of the high-fidelity code.
![Development of Multi-Level Reduced Order Modeling Methodology
Mohammad G. Abdo and Hany S. Abdel-Khalik
School of Nuclear Engineering, Purdue University
400 Central Drive, Purdue Campus, NUCL Bldg., West Lafayette, IN 47906
abdo@purdue.edu; abdelkhalik@purdue.edu
INTRODUCTION
Over the past decade, the nuclear engineering
community in the United States and abroad has been heavily
investing in the development and implementation of
advanced science-based (i.e., employing fine-mesh, first-
principles, tightly coupled multi-physics) modeling and
simulation of nuclear reactor systems. Although these
models are expected to improve the understanding of the
reactors behavior, their performance is expected to be
handicapped by the enormous computational cost required
for their execution, especially when computationally
intensive analyses are of interest such as uncertainty
quantification and design optimization. To address this
challenge, reduced order modeling (ROM) techniques [1, 2]
development has been underway to help realize the goals of
advanced science-based modeling and simulation.
Past work has demonstrated a number of algorithms
that can render reduction for reactor physics calculations
[3]. The general outcome of this research is that one may
render reduction for a high fidelity model by executing such
model a number of times r (referred to as the effective
dimensionality of the model and is typically in the order of
few hundreds for reactor models), which is several orders of
magnitude smaller than the original model dimensionality m
which measures either the size of the state space (i.e., flux),
the parameters (e.g., cross-sections), or responses (e.g., pin
power distribution), all expected to be very large in
dimensionality. Notwithstanding this significant reduction,
the execution of high fidelity codes few hundred times is
still impractical even on leadership supercomputers. We
therefore present a new development in ROM, referred to as
multi-level ROM, which allows one to extract the effective
dimensionality by executing the high fidelity model in a
small sub-domain of the overall problem domain, i.e., pin
cell or 2D lattice vs. whole core. We show that this
approach is as capable as the previously developed ROM
approach in capturing the errors resulting from the reduction
and establishing realistic upper-bounds on their magnitude
over the expected range of variations for the high fidelity
model.
DESCRIPTION OF THE ACTUAL WORK
Unlike conventional surrogate construction techniques,
or low order physics approximations, ROM offers a
rigorous approach by which the effective dimensionality of
the physics model is determined without altering the
formulation of the high fidelity physics model. The idea is
to assume that the state variables, the input parameters, and
the output responses, are varying over lower dimensional
manifolds (referred to as active subspaces). See below for
the mathematical description of ROM techniques.
Any type of reduction introduces errors due to the
discarding of non-influential components. Capturing these
errors, and more importantly upper-bounding them, is very
crucial to lend credibility to the reduction process. Past
work has arrived at two important results. The first result
deals with the straightforward application of ROM to the
high fidelity model, wherein the active subspace is
constructed based on few hundred executions of the high
fidelity model. In this case, upper-bounds on the reduction
errors can be established with sufficiently high probability,
i.e., 0.99999. We refer to this active subspace as the ideal
active subspace, since it is demonstrated to have the
smallest size as compared to the subspaces constructed in
this work. The second result of past work has proved that
one could develop an upper-bound on the ROM error for
any user-defined active subspace. Of course, if the subspace
employed is a bad approximation to the ideal active
subspace, the error bounds would be too high. We employ
this second result in this summary by employing a physics-
informed application of the high fidelity model to sub-
regions of the problem domain to extract the active
subspace. This approach is referred to as multi-level ROM,
or MLROM. The physics of the problem will be employed
to inform our choice of the sub-domain(s) used to represent
the full order model. In particular, we consider a 2D lattice
model depleted to 60 GWd/MTU to represent the full high
fidelity model. The goal is to employ few static (i.e.,
without depletion) pin cell models to identify the active
subspace. Error bounds will be constructed using the pin-
cell-determined active subspace, which will be verified
numerically by comparing against the non-reduced (i.e., full
order) simulation of the lattice model. Static evaluation of
the flux for a pin cell model few hundred times is
computationally inexpensive. In future work, this idea will
be applied to core-wide calculations, wherein some
representative lattices may also be needed to capture the
core-wide active subspace. This will be investigated in
future work.](https://image.slidesharecdn.com/multilevelromanssummer2015revmarch23-150709235227-lva1-app6892/85/Development-of-Multi-level-Reduced-Order-MOdeling-Methodology-2-320.jpg)
![MATHEMATICAL DESCRIPTION OF MLROM
Consider a reactor physics model, abstractly defined as:
y f x (1)
where xm
x are reactor physics parameters, e.g., cross-
sections, ym
y are reactor responses of interest e.g.,
eigenvalue, peak clad temperature, reaction rates, etc., and
mx and my are the numbers of parameters and responses,
respectively.
The ROM replaces the original simulation with a
reduced order representation f such that:
y xf x f x T T (2)
where both Tx and Ty are rank-deficient linear mappings for
parameters and response spaces, respectively, described by:
z zm mT
z z z
T Q Q for ,z x y
The effective dimensionalities of these operators, i.e.
dim R zz mrT - with R being the range of the operator
Tz - are typically much smaller than the original dimensions
of the parameters and responses spaces, i.e.,
max( , ) min( , )x ym m x yr r m m . The columns of the matrix
x mx
m r
x
Q define the range of all possible input parameter
perturbations that have dominant impact on the response of
interest, whereas the matrix
y my
m r
y
Q define the range of
all possible response variations resulting from all possible
parameter variations. Depending on the number of
parameters and responses, one may elect to render a single
reduction in the parameter or response spaces, or
simultaneous reduction in both spaces. A number of
algorithms have been developed to achieve that; the reader
may consult earlier work for more details on the
construction of the Q matrices [3, 4].
To ensure the ROM model is credible, one must bound
the errors resulting from discarding components in the input
parameter space and/or the response space. To achieve that,
we define the following error matrix E which calculates the
exact reduction errors using a number of samples. The ijth
element of E defines the error in the ith
response recorded in
the jth
random sample:
,: ,: ( )
[ ]
T T
i j y y i x x j
ij
i j
f x i i f x
f x
Q Q Q Q
E (3)
The elements of this error matrix can be used to
calculate an upper-bound on the error via the following
theoretical result due to Dixon [5]:
2
2
1
1
01,2,
max 1 ( ) ,
i s
xi s
x pdf t dt
E E
(4)
where the multiplier 1 x is arbitrary. Recent work has
determined a numerical value of 1.0164 for this multiplier
which ensures that the bound is not too conservative. The
straightforward application of Dixon theory employs a
multiplier of approximately 8. Details on this aspect of the
error bound determination may be found in the following
references [6, 7].
In this work we assume that no reduction is done in the
response space, i.e., y T I is the identity matrix, while xT
is determined using gradient-based ROM approach applied
to a pin cell model, for more details about this approach
reader can consult the work in [3]
NUMERICAL TESTS AND RESULTS
For this demonstration, we employ a benchmark lattice
model for the Peach Bottom Atomic Power Station Unit 2
(PB-2), which is a 1112 MWe Boiling Water Reactor
(BWR) manufactured by General Electric with the fuel rods
arranged over a 7x7 grid. This benchmark is designed by
OECD/NEA and documented in [8]. Given the requirement
to have flat pin power distribution, several pin cell design
(different enrichments and gad content) are typically
employed in BWR lattices.
The 2.93% enriched UO2 with 3% gadolinium pin cell
is first depleted to 30 GWd/MTU. The resulting
composition is used as the reference model for constructing
the subspace. This pin is assumed to be the most
representative of all pins in the lattice. Note here that our
goal is to identify the dominant cross-sections for all the
pins, hence this step requires familiarity with the model.
After the subspace is constructed, we employ it to model
other fuel pins to establish whether our initial assumption of
representativity is adequate. If the assumption is adequate,
we move to the next level, and test its adequacy to represent
the entire lattice; if not, other pins are added to construct a
more representative subspace.
The nominal dimension mx of the parameter subspace
for the reference pin cell is (93 fuel nuclides + 15 clad
nuclides + 2 moderator nuclides + 1 gap nuclide) * 3
reactions * 44 energy groups =14,652 parameters. The
effective dimensionality is selected to be 600. Note here that
the error bounds are expected to vary from one response to
the next because of two reasons. First, depending on the
quality of the active subspace, different responses are
expected to be captured to different degrees of accuracy.
Second, because ROM attempts to identify the most
influential components, non-influential responses such as
the very fast and very thermal groups are expected to have
higher relative errors than the rest of the groups.
Figures 1 through 8 illustrate how the subspace
extracted from the low-fidelity model behaves when used
with different pin cells. Even-numbered figures display the
errors in the flux range of (1.85 – 2.35 MeV), whereas the
odd-numbered figures are for the range (0.625 – 1.0 eV).
Figures 1 through 4 show how the subspace behaves if used
to identify the parameters of a 1.33% enriched UO2 pin cell,
whereas figures 5 through 8 are for a 2.93% enriched UO2](https://image.slidesharecdn.com/multilevelromanssummer2015revmarch23-150709235227-lva1-app6892/85/Development-of-Multi-level-Reduced-Order-MOdeling-Methodology-3-320.jpg)
Development of Multi-level Reduced Order MOdeling Methodology
- 2. Development of Multi-Level Reduced Order Modeling Methodology Mohammad G. Abdo and Hany S. Abdel-Khalik School of Nuclear Engineering, Purdue University 400 Central Drive, Purdue Campus, NUCL Bldg., West Lafayette, IN 47906 abdo@purdue.edu; abdelkhalik@purdue.edu INTRODUCTION Over the past decade, the nuclear engineering community in the United States and abroad has been heavily investing in the development and implementation of advanced science-based (i.e., employing fine-mesh, first- principles, tightly coupled multi-physics) modeling and simulation of nuclear reactor systems. Although these models are expected to improve the understanding of the reactors behavior, their performance is expected to be handicapped by the enormous computational cost required for their execution, especially when computationally intensive analyses are of interest such as uncertainty quantification and design optimization. To address this challenge, reduced order modeling (ROM) techniques [1, 2] development has been underway to help realize the goals of advanced science-based modeling and simulation. Past work has demonstrated a number of algorithms that can render reduction for reactor physics calculations [3]. The general outcome of this research is that one may render reduction for a high fidelity model by executing such model a number of times r (referred to as the effective dimensionality of the model and is typically in the order of few hundreds for reactor models), which is several orders of magnitude smaller than the original model dimensionality m which measures either the size of the state space (i.e., flux), the parameters (e.g., cross-sections), or responses (e.g., pin power distribution), all expected to be very large in dimensionality. Notwithstanding this significant reduction, the execution of high fidelity codes few hundred times is still impractical even on leadership supercomputers. We therefore present a new development in ROM, referred to as multi-level ROM, which allows one to extract the effective dimensionality by executing the high fidelity model in a small sub-domain of the overall problem domain, i.e., pin cell or 2D lattice vs. whole core. We show that this approach is as capable as the previously developed ROM approach in capturing the errors resulting from the reduction and establishing realistic upper-bounds on their magnitude over the expected range of variations for the high fidelity model. DESCRIPTION OF THE ACTUAL WORK Unlike conventional surrogate construction techniques, or low order physics approximations, ROM offers a rigorous approach by which the effective dimensionality of the physics model is determined without altering the formulation of the high fidelity physics model. The idea is to assume that the state variables, the input parameters, and the output responses, are varying over lower dimensional manifolds (referred to as active subspaces). See below for the mathematical description of ROM techniques. Any type of reduction introduces errors due to the discarding of non-influential components. Capturing these errors, and more importantly upper-bounding them, is very crucial to lend credibility to the reduction process. Past work has arrived at two important results. The first result deals with the straightforward application of ROM to the high fidelity model, wherein the active subspace is constructed based on few hundred executions of the high fidelity model. In this case, upper-bounds on the reduction errors can be established with sufficiently high probability, i.e., 0.99999. We refer to this active subspace as the ideal active subspace, since it is demonstrated to have the smallest size as compared to the subspaces constructed in this work. The second result of past work has proved that one could develop an upper-bound on the ROM error for any user-defined active subspace. Of course, if the subspace employed is a bad approximation to the ideal active subspace, the error bounds would be too high. We employ this second result in this summary by employing a physics- informed application of the high fidelity model to sub- regions of the problem domain to extract the active subspace. This approach is referred to as multi-level ROM, or MLROM. The physics of the problem will be employed to inform our choice of the sub-domain(s) used to represent the full order model. In particular, we consider a 2D lattice model depleted to 60 GWd/MTU to represent the full high fidelity model. The goal is to employ few static (i.e., without depletion) pin cell models to identify the active subspace. Error bounds will be constructed using the pin- cell-determined active subspace, which will be verified numerically by comparing against the non-reduced (i.e., full order) simulation of the lattice model. Static evaluation of the flux for a pin cell model few hundred times is computationally inexpensive. In future work, this idea will be applied to core-wide calculations, wherein some representative lattices may also be needed to capture the core-wide active subspace. This will be investigated in future work.
- 3. MATHEMATICAL DESCRIPTION OF MLROM Consider a reactor physics model, abstractly defined as: y f x (1) where xm x are reactor physics parameters, e.g., cross- sections, ym y are reactor responses of interest e.g., eigenvalue, peak clad temperature, reaction rates, etc., and mx and my are the numbers of parameters and responses, respectively. The ROM replaces the original simulation with a reduced order representation f such that: y xf x f x T T (2) where both Tx and Ty are rank-deficient linear mappings for parameters and response spaces, respectively, described by: z zm mT z z z T Q Q for ,z x y The effective dimensionalities of these operators, i.e. dim R zz mrT - with R being the range of the operator Tz - are typically much smaller than the original dimensions of the parameters and responses spaces, i.e., max( , ) min( , )x ym m x yr r m m . The columns of the matrix x mx m r x Q define the range of all possible input parameter perturbations that have dominant impact on the response of interest, whereas the matrix y my m r y Q define the range of all possible response variations resulting from all possible parameter variations. Depending on the number of parameters and responses, one may elect to render a single reduction in the parameter or response spaces, or simultaneous reduction in both spaces. A number of algorithms have been developed to achieve that; the reader may consult earlier work for more details on the construction of the Q matrices [3, 4]. To ensure the ROM model is credible, one must bound the errors resulting from discarding components in the input parameter space and/or the response space. To achieve that, we define the following error matrix E which calculates the exact reduction errors using a number of samples. The ijth element of E defines the error in the ith response recorded in the jth random sample: ,: ,: ( ) [ ] T T i j y y i x x j ij i j f x i i f x f x Q Q Q Q E (3) The elements of this error matrix can be used to calculate an upper-bound on the error via the following theoretical result due to Dixon [5]: 2 2 1 1 01,2, max 1 ( ) , i s xi s x pdf t dt E E (4) where the multiplier 1 x is arbitrary. Recent work has determined a numerical value of 1.0164 for this multiplier which ensures that the bound is not too conservative. The straightforward application of Dixon theory employs a multiplier of approximately 8. Details on this aspect of the error bound determination may be found in the following references [6, 7]. In this work we assume that no reduction is done in the response space, i.e., y T I is the identity matrix, while xT is determined using gradient-based ROM approach applied to a pin cell model, for more details about this approach reader can consult the work in [3] NUMERICAL TESTS AND RESULTS For this demonstration, we employ a benchmark lattice model for the Peach Bottom Atomic Power Station Unit 2 (PB-2), which is a 1112 MWe Boiling Water Reactor (BWR) manufactured by General Electric with the fuel rods arranged over a 7x7 grid. This benchmark is designed by OECD/NEA and documented in [8]. Given the requirement to have flat pin power distribution, several pin cell design (different enrichments and gad content) are typically employed in BWR lattices. The 2.93% enriched UO2 with 3% gadolinium pin cell is first depleted to 30 GWd/MTU. The resulting composition is used as the reference model for constructing the subspace. This pin is assumed to be the most representative of all pins in the lattice. Note here that our goal is to identify the dominant cross-sections for all the pins, hence this step requires familiarity with the model. After the subspace is constructed, we employ it to model other fuel pins to establish whether our initial assumption of representativity is adequate. If the assumption is adequate, we move to the next level, and test its adequacy to represent the entire lattice; if not, other pins are added to construct a more representative subspace. The nominal dimension mx of the parameter subspace for the reference pin cell is (93 fuel nuclides + 15 clad nuclides + 2 moderator nuclides + 1 gap nuclide) * 3 reactions * 44 energy groups =14,652 parameters. The effective dimensionality is selected to be 600. Note here that the error bounds are expected to vary from one response to the next because of two reasons. First, depending on the quality of the active subspace, different responses are expected to be captured to different degrees of accuracy. Second, because ROM attempts to identify the most influential components, non-influential responses such as the very fast and very thermal groups are expected to have higher relative errors than the rest of the groups. Figures 1 through 8 illustrate how the subspace extracted from the low-fidelity model behaves when used with different pin cells. Even-numbered figures display the errors in the flux range of (1.85 – 2.35 MeV), whereas the odd-numbered figures are for the range (0.625 – 1.0 eV). Figures 1 through 4 show how the subspace behaves if used to identify the parameters of a 1.33% enriched UO2 pin cell, whereas figures 5 through 8 are for a 2.93% enriched UO2
- 4. pin cell. Figures 1 and 2 show the actual error vs. the error bound due to parameter reduction in a 1.33% enriched UO2 pin cell after depleting to 15 GWd/MTU. In both figures the red dots represent failed cases where the blue ones represent the success, i.e., when the error bound is exceeded by the actual error. All the cases showed a failure probability of less than 0.1 as the theory predicts. The red solid 45-degree line separates the failure and success regions. In this demonstration, we picked a low probability of success, i.e., 0.9, in order to realize some failure. In practice however, the success probability is set to at least 0.99999. Fig. 1. Fast Flux Error (15 GWd/MTU ). The previous figure shows that the maximum actual error in the fast flux range is 0.12%. Fig. 2. Thermal Flux Error (15 GWd/MTU). Fig. 3. Fast Flux Error (45 GWd/MTU). Fig. 4. Thermal Flux Error (45 GWd/MTU). Figure 2 shows similar results for the thermal range. Figures 3 and 4 examine the active subspace applied on the same pin cell if the initial composition resulted from a 45 GWd/MTU depletion. The previous figures show that the maximum error in the specified fast and thermal ranges are 0.28% and 0.42%, respectively. Figures 5 and 6 show the fast and thermal flux errors for the 2.93% enriched UO2 pincell after depleting to 15 GWd/MTU. Fig. 5. Fast Flux Error (15GWd/MTU). Fig. 6. Thermal Flux Errors (15 GWd/MTU). Maximum errors in fast and thermal flux are found to be 0.3% and 0.33% respectively. Finally, figures 7 and 8 plot the errors if the pin cell is initially depleted to 45 GWd/MTU.
- 5. Fig. 7. Fast Flux Errors (45 GWd/MTU). Again the previous figure shows that constraining the cross sections to the active space extracted from the reference pin cell model resulted in an error that does not exceed 0.3% for the specified fast range of (1.85 – 2.35 MeV). Fig. 8. Thermal Flux Errors (45 GWd/MTU). The thermal range errors follow the same pattern and recorded a maximum value of 0.3%. CONCLUSIONS This exploratory studies investigate a new idea to construct the active subspace for a model that is computationally expensive to execute. The basic idea is to employ the physics in a sub-domain, akin to lattice physics calculations, and let the reduction theory determine the error bounds resulting from this application to other pin cell models. Initial results indicate that the idea is sound for the lattice model employed representing an LWR reactor. The next step will be to apply this idea to the lattice level. We would like to point out that while the idea here is similar to standard homogenization theory (at least in spirit, since no actual homogenization is applied), we are able to calculate error bounds on the active subspace employed, which is not possible with existing homogenization theory techniques. If successful, this approach could help realize the potential of employing high fidelity simulation tools, currently under development in many institutions around the country, in a practical manner that can benefit the end-users, i.e., nuclear practitioners. ACKNOWLEDGMENT The first author would like to acknowledge the support received from the department of nuclear engineering at North Carolina State University to complete this work in support of his PhD. REFERENCES 1. W. H. A. SCHILDERS, H. A. VAN DER VORST, J: ROMMES (eds.), “Model Ordwer Reductin: Theory, Research Aspects and Applications,” Springer: Berlin, 2008. 2. ANTOULAS AC, SORESEN DC, GUGERCIN S. “A Servay of Model Reduction methods for Large Scale Systems,” Journal of Contemporary Mathematics 2001; 280:193-219. 3. YOUNGSUK BANG, JASON HITE, AND HANY S. ABDEL-KHALIK, “Hybrid Reduced Order Modeling Applied To Nonlinear Models,” IJNME; 91: 929–949 (2012). 4. CONGJIAN WANG, HANY S. ABDEL-KHALIK, “Exact-To-Precision Generalized Perturbation Theory For Eigen Value Problems,” Nuclear Engineering and Design, 256, pp. 130-140, (2013). 5. JOHN D. DIXON, “Estimating Extremal Eigenvalues And Condition Numbers Of Matrices,” SIAM 1983; 20(2): 812–814. 6. MOHAMMAD ABDO AND HANY S. ABDEL- KHALIK, “Further Investigation of Error Bounds for Reduced Order Modeling,” Proceedings of Int. Conf. Math and Comp., Nashville, TN 2015. 7. MOHAMMAD ABDO AND HANY S. ABDEL- KHALIK, “Probabilistic Error Bounds for Reduced Order Modeling,” Trans. Am. Nucl. Soc., 110, 2014. 8. BEDIRHAN AKDENIZ, KOSTADIN N. IVANOV, “Boiling Water Reactor Turbine Trip (TT) Benchmark; Volume III: Summary Results of Exercise 2,” NEA/NSC/DOC(2006)23. 9. HANY S. ABDEL-KHALIK, ET AL., “Overview of Hybrid Subspace Methods for Uncertainty Quantification and Sensitivity Analysis,” Annals of Nuclear Energy, 52, pp.28-46 (2013). A Tutorial on Applications of dimensionality reduction and function approximation. 10. SCALE: A Comprehensive Modeling and Simulation Suite for Safety Analysis and Design, ORNL/TM- 2005/39, Version 6.1, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 2011.