I am a professor of nuclear engineering whose research interests include the development of numerical methods to solve the equations describing steady-state and transient behavior of nuclear reactors including neutron transport, gamma transport, and multiphysics phenomena.
MC4 is a detector simulation program combining a vectorized ray-tracing algorithm with a vectoriz... more MC4 is a detector simulation program combining a vectorized ray-tracing algorithm with a vectorized version of the electromagnetic interaction routines from GEANT3. The implementation of ray tracing is able to represent moderately complex geometries such as single calorimeter modules or test-beam situations. Results from MC4 are compared with EGS4 simulations and with experimental results. Timing results are given for scalar machines and on a vector supercomputer. Production applications and applications to future versions of the GEANT code are discussed. Partially funded by the US Department of Energy through contract number DE-FCO5-85ER250000. 2 Current address: EVEREX Co.
Kernel Density Estimators (KDEs) are a non-parametric density estimation technique that has recen... more Kernel Density Estimators (KDEs) are a non-parametric density estimation technique that has recently been applied to Monte Carlo radiation transport simulations . Kernel density estimators are an alternative to histogram tallies for obtaining global solutions in Monte Carlo tallies. With KDEs, a single event, either a collision or particle track, can contribute to the score at multiple tally points with the uncertainty at those points being independent of the desired resolution of the solution. Thus, KDEs show potential for obtaining estimates of a global solution with reduced variance when compared to a histogram. Previously, KDEs have been applied to neutronics for one-group reactor physics problems [1] and fixed source shielding applications . However, little work was done to obtain reaction rates using KDEs. Previously, the Mean Free Path (MFP) based KDE was introduced that is capable of obtaining accurate estimates of reaction rates for reactor physics problems in 1-D slab geometries in continuous energy and 2-D one-group problems with linear material interfaces. However, the MFP KDE was not extended to 2-D geometries with non-planar surfaces . This paper introduces a new form of the MFP KDE that is capable of handling general geometries. Furthermore, extending the MFP KDE to 2-D problems in continuous energy introduces inaccuracies to the solution. An ad-hoc solution to these inaccuracies is introduced that produces errors smaller than 4% at material interfaces. Additionally, While KDEs produce smoother results compared to histograms, it comes at a cost of increased computation time. For every particle event, a kernel function must be evaluated for every tally point within the support range of the event. Furthermore, tallying to points in materials outside of where the particle event occurred requires the look up of additional cross section information. Both of these facts can make the KDE tally routine the most expensive portion of the Monte Carlo simulation. Since the KDE requires the calculation of multiple quantities for every particle event, it is well suited for computation on a Graphics Processing Unit (GPU). In an attempt to reduce run times, the KDE tally is exported to the GPU during the transport process. The KDE is applied to tallies in two 2-D pincell problems as well as two quarter-assembly problems. Speedups are problem dependent, and range between 1.6 and 13.8 for the problems studied in this paper.
In order to efficiently use new features of supercomputers, production codes, usually written 10 ... more In order to efficiently use new features of supercomputers, production codes, usually written 10 -20 years ago, must be tailored for modern computer architectures. We have chosen to optimize the CPM-2 code, a production reactor assembly code based on the collision probability transport method. Substantional speedups in the execution times were obtained with the parallel/vector version of the CPM-2 code. In addition, we have developed a new transfer probability method, which removes some of the modelling limitations of the collision probability method encoded in the CPM-2 code, and can fully utilize parallel/vector architecture of a mulfiprocessor IBM 3090.
Journal of Quantitative Spectroscopy & Radiative Transfer, Feb 1, 1990
Monte Carlo method is described for computing the backscattering of radiation from a sphere situa... more Monte Carlo method is described for computing the backscattering of radiation from a sphere situated above an emitting plane. Numerical results are given and compared with analytic bounds for this problem which were previously reported in this journal.
Most particle transport Monte Carlo codes in use today are based on the "history-based" algorithm... more Most particle transport Monte Carlo codes in use today are based on the "history-based" algorithm, wherein one particle history at a time is simulated. Unfortunately, the "history-based" approach (present in all Monte Carlo codes until recent years) is inherently scalar and cannot be vectorized. In particular, the history-based algorithm cannot take advantage of vector architectures, which characterize the largest and fastest computers at the Current time, vector supercomputers such as the Cray X/MP or IBM 3090/600. However, substantial progress has been made in recent years in developing and implementing a vectorized Monte Carlo algorithm. This algorithm follows portions of many particle histories at the same time and forms the basis for all successful vectorized Monte Carlo codes that are in use today. This paper describes the basic vectorized algorithm along with descriptions of several variations that have been developed by different researchers for specific applications. These applications have been mainly in the areas of neutron transport in nuclear reactor and shielding analysis and photon transport in fusion plasmas. The relative merits of the various approach schemes will be discussed and the present status of known vectorization efforts will be summarized along with available timing results, including results from the successful vectorization of 3-D general geometry, Continuous energy Monte Carlo.
The finite element response matrix method has been applied to the solution of the neutron transpo... more The finite element response matrix method has been applied to the solution of the neutron transport equation. This method has previously been applied to the neutron diffusion equation for coarse mesh reactor analysis with excellent results. As with the diffusion equation implementation, the transport method employs a local-global projection technique to allow treatment of internal heterogeneities that would normally not be resolved by the coarse global mesh that is needed for computational efficiency. However, since the transport equation includes the angular domain, the local~lobal projection technique has been extended to angle as well as space. Since the response matrices do not depend on the multiplication factor, a conventional fission source iteration method is utilized for criticality problems. The method has been applied to the one-dimensional and twodimensional neutron transport equations. For one-dimensional geometries, the local global projection method yields excellent results, indicating the potential of this approach as a viable coarse mesh transport method. Numerical results are presented for several one-dimensional configurations that were analyzed with varying choices of local and global meshes in the spatial domain. Preliminary results with two-dimensional applications indicate that computational times may be an order of magnitude faster than with the conventional finite element solution of the two-dimensional transport equation.
Conventional resonance self-shielding methods are primarily based on equivalence theory, which al... more Conventional resonance self-shielding methods are primarily based on equivalence theory, which allows the resonance integral (RI) table generated by homogeneous media to be useful for heterogeneous calculation. In the past two decades, a new RI table based directly on heterogeneous calculation has been developed and proved to be effective in the self-shielding calculation. This note compares the homogeneous and heterogeneous RI tables on the basis of derivation and numerical results. The limitations of the two RI tables and their applicability to the embedded self-shielding method (ESSM) are discussed. Conventional resonance theory breaks the solution roadmap into two steps: (1) for homogeneous media, tabulate resonance integral (RI) of a resonance isotope against background cross section; (2) correlate heterogeneous system with homogeneous system using equivalence theory [1] so that the table generated for homogeneous media can be utilized for
This report was prepared as an account of work sponsored by the United States Government. Neither... more This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product or process disclosed or represents that its use would not infl'inge privately-owned rights.
The time-dependent behavior of the energy spectrum in neutron transport was investigated with a f... more The time-dependent behavior of the energy spectrum in neutron transport was investigated with a formulation, based on continuous-time Markov processes, for computing α eigenvalues and eigenvectors in an infinite medium. For this, a research Monte Carlo code called "TORTE" (To Obtain Real Time Eigenvalues) was created and used to estimate elements of a transition rate matrix. TORTE is capable of using both multigroup and continuous-energy nuclear data, and verification was performed. Eigenvalue spectra for infinite
The development of advanced computers with special capabilities for vectorized or parallel calcul... more The development of advanced computers with special capabilities for vectorized or parallel calculations demands the development of new calculational methods. The very nature of the Monte Carlo process precludes direct conversion of old (scalar) codes to the new machines. Instead, major changes in global algorithms and careful selection of compatible physics treatments are required. Recent results for Monte Carlo in multigroup shielding applications and in continuous-energy reactor lattice analysis have demonstrated that Monte Carlo methods can be successfully vectorized. The significant effort required for stylized coding and major algorithmic changes is worthwhile, and significant gains in computational efficiency are realized. Speedups of at least twenty to forty times faster than CDC-7600 scalar calculations have been achieved on the CYBER-205 without sacrificing the accuracy of standard Monte Carlo methods. Speedups of this magnitude provide reductions in statistical uncertainties for a given amount of computing time, permit more detailed and realistic problems to be analyzed, and make the Monte Carlo method more accessible to nuclear analysts. Following overviews of the Monte Carlo method for particle transport analysis and of vector computer hardware and software characteristics, both general and specific aspects of the vectorization of Monte Carlo are discussed. Finally, numerical results obtained from vectorized Monte Carlo codes run on the CYBER-205 are presented.
This report was prepared as an account of work sponsored by the United States Government. Neither... more This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product or process disclosed or represents that its use would not infl'inge privately-owned rights.
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Oct 26, 2011
The standard discrete thermal neutron S(a,~) scattering treatment in MCNP5 is compared with a con... more The standard discrete thermal neutron S(a,~) scattering treatment in MCNP5 is compared with a continuous S( a, ~) scattering treatment using a criticality suite of 119 benchmark cases and ENDF IB-VILO nuclear data. In the analysis, six bound isotopes are corisidered: beryllium metal, graphite, hydrogen in water, hydrogen in polyethylene, beryllium in beryllium oxide and oxygen in beryllium oxide. Overall, there are only small changes in the eigenvalue (k eff ) between discrete and continuous treatments. In the comparison of 64 cases that utilize S(a,~) scattering, 62 agreed at the 95% confidence level, and the 2 cases with differences larger than 3 (J agreed exactly when more neutrons were run in the calculations. The results indicate that the changes in eigenvalue between continuous and discrete treatments are random, small, and well within the uncertainty of measured data for reactor criticality experiments.
We describe the initial experience and results from implementing a fission matrix capability into... more We describe the initial experience and results from implementing a fission matrix capability into the MCNP Monte Carlo code. The fission matrix is obtained at essentially no cost during the normal simulation for criticality calculations. It can be used to provide estimates of the fundamental mode fission distribution, the dominance ratio, the eigenvalue spectrum, and higher mode spatial eigenfunctions. It can also be used to accelerate the convergence of the power method iterations and to provide basis functions for higher-order perturbation theory. Past difficulties and limitations of the fission matrix approach are overcome with a new sparse representation of the matrix, permitting much larger and more accurate fission matrix representations. The new fission matrix capabilities provide a significant advance in the state-of-the-art for Monte Carlo criticality calculations.! LA-UR-12-xxxxx Monte Carlo Criticality Calculations" Monte Carlo K-effective Calculation" 1. Start with fission source & eigenvalue guess" 2. Repeat until converged:! • Simulate neutrons, save fission sites for next cycle" • Calculate k-eff, renormalize source" 3. Continue iterating & tally quantities of interest! 4! LA-UR-12-xxxxx
MC4 is a detector simulation program combining a vectorized ray-tracing algorithm with a vectoriz... more MC4 is a detector simulation program combining a vectorized ray-tracing algorithm with a vectorized version of the electromagnetic interaction routines from GEANT3. The implementation of ray tracing is able to represent moderately complex geometries such as single calorimeter modules or test-beam situations. Results from MC4 are compared with EGS4 simulations and with experimental results. Timing results are given for scalar machines and on a vector supercomputer. Production applications and applications to future versions of the GEANT code are discussed. Partially funded by the US Department of Energy through contract number DE-FCO5-85ER250000. 2 Current address: EVEREX Co.
Kernel Density Estimators (KDEs) are a non-parametric density estimation technique that has recen... more Kernel Density Estimators (KDEs) are a non-parametric density estimation technique that has recently been applied to Monte Carlo radiation transport simulations . Kernel density estimators are an alternative to histogram tallies for obtaining global solutions in Monte Carlo tallies. With KDEs, a single event, either a collision or particle track, can contribute to the score at multiple tally points with the uncertainty at those points being independent of the desired resolution of the solution. Thus, KDEs show potential for obtaining estimates of a global solution with reduced variance when compared to a histogram. Previously, KDEs have been applied to neutronics for one-group reactor physics problems [1] and fixed source shielding applications . However, little work was done to obtain reaction rates using KDEs. Previously, the Mean Free Path (MFP) based KDE was introduced that is capable of obtaining accurate estimates of reaction rates for reactor physics problems in 1-D slab geometries in continuous energy and 2-D one-group problems with linear material interfaces. However, the MFP KDE was not extended to 2-D geometries with non-planar surfaces . This paper introduces a new form of the MFP KDE that is capable of handling general geometries. Furthermore, extending the MFP KDE to 2-D problems in continuous energy introduces inaccuracies to the solution. An ad-hoc solution to these inaccuracies is introduced that produces errors smaller than 4% at material interfaces. Additionally, While KDEs produce smoother results compared to histograms, it comes at a cost of increased computation time. For every particle event, a kernel function must be evaluated for every tally point within the support range of the event. Furthermore, tallying to points in materials outside of where the particle event occurred requires the look up of additional cross section information. Both of these facts can make the KDE tally routine the most expensive portion of the Monte Carlo simulation. Since the KDE requires the calculation of multiple quantities for every particle event, it is well suited for computation on a Graphics Processing Unit (GPU). In an attempt to reduce run times, the KDE tally is exported to the GPU during the transport process. The KDE is applied to tallies in two 2-D pincell problems as well as two quarter-assembly problems. Speedups are problem dependent, and range between 1.6 and 13.8 for the problems studied in this paper.
In order to efficiently use new features of supercomputers, production codes, usually written 10 ... more In order to efficiently use new features of supercomputers, production codes, usually written 10 -20 years ago, must be tailored for modern computer architectures. We have chosen to optimize the CPM-2 code, a production reactor assembly code based on the collision probability transport method. Substantional speedups in the execution times were obtained with the parallel/vector version of the CPM-2 code. In addition, we have developed a new transfer probability method, which removes some of the modelling limitations of the collision probability method encoded in the CPM-2 code, and can fully utilize parallel/vector architecture of a mulfiprocessor IBM 3090.
Journal of Quantitative Spectroscopy & Radiative Transfer, Feb 1, 1990
Monte Carlo method is described for computing the backscattering of radiation from a sphere situa... more Monte Carlo method is described for computing the backscattering of radiation from a sphere situated above an emitting plane. Numerical results are given and compared with analytic bounds for this problem which were previously reported in this journal.
Most particle transport Monte Carlo codes in use today are based on the "history-based" algorithm... more Most particle transport Monte Carlo codes in use today are based on the "history-based" algorithm, wherein one particle history at a time is simulated. Unfortunately, the "history-based" approach (present in all Monte Carlo codes until recent years) is inherently scalar and cannot be vectorized. In particular, the history-based algorithm cannot take advantage of vector architectures, which characterize the largest and fastest computers at the Current time, vector supercomputers such as the Cray X/MP or IBM 3090/600. However, substantial progress has been made in recent years in developing and implementing a vectorized Monte Carlo algorithm. This algorithm follows portions of many particle histories at the same time and forms the basis for all successful vectorized Monte Carlo codes that are in use today. This paper describes the basic vectorized algorithm along with descriptions of several variations that have been developed by different researchers for specific applications. These applications have been mainly in the areas of neutron transport in nuclear reactor and shielding analysis and photon transport in fusion plasmas. The relative merits of the various approach schemes will be discussed and the present status of known vectorization efforts will be summarized along with available timing results, including results from the successful vectorization of 3-D general geometry, Continuous energy Monte Carlo.
The finite element response matrix method has been applied to the solution of the neutron transpo... more The finite element response matrix method has been applied to the solution of the neutron transport equation. This method has previously been applied to the neutron diffusion equation for coarse mesh reactor analysis with excellent results. As with the diffusion equation implementation, the transport method employs a local-global projection technique to allow treatment of internal heterogeneities that would normally not be resolved by the coarse global mesh that is needed for computational efficiency. However, since the transport equation includes the angular domain, the local~lobal projection technique has been extended to angle as well as space. Since the response matrices do not depend on the multiplication factor, a conventional fission source iteration method is utilized for criticality problems. The method has been applied to the one-dimensional and twodimensional neutron transport equations. For one-dimensional geometries, the local global projection method yields excellent results, indicating the potential of this approach as a viable coarse mesh transport method. Numerical results are presented for several one-dimensional configurations that were analyzed with varying choices of local and global meshes in the spatial domain. Preliminary results with two-dimensional applications indicate that computational times may be an order of magnitude faster than with the conventional finite element solution of the two-dimensional transport equation.
Conventional resonance self-shielding methods are primarily based on equivalence theory, which al... more Conventional resonance self-shielding methods are primarily based on equivalence theory, which allows the resonance integral (RI) table generated by homogeneous media to be useful for heterogeneous calculation. In the past two decades, a new RI table based directly on heterogeneous calculation has been developed and proved to be effective in the self-shielding calculation. This note compares the homogeneous and heterogeneous RI tables on the basis of derivation and numerical results. The limitations of the two RI tables and their applicability to the embedded self-shielding method (ESSM) are discussed. Conventional resonance theory breaks the solution roadmap into two steps: (1) for homogeneous media, tabulate resonance integral (RI) of a resonance isotope against background cross section; (2) correlate heterogeneous system with homogeneous system using equivalence theory [1] so that the table generated for homogeneous media can be utilized for
This report was prepared as an account of work sponsored by the United States Government. Neither... more This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product or process disclosed or represents that its use would not infl'inge privately-owned rights.
The time-dependent behavior of the energy spectrum in neutron transport was investigated with a f... more The time-dependent behavior of the energy spectrum in neutron transport was investigated with a formulation, based on continuous-time Markov processes, for computing α eigenvalues and eigenvectors in an infinite medium. For this, a research Monte Carlo code called "TORTE" (To Obtain Real Time Eigenvalues) was created and used to estimate elements of a transition rate matrix. TORTE is capable of using both multigroup and continuous-energy nuclear data, and verification was performed. Eigenvalue spectra for infinite
The development of advanced computers with special capabilities for vectorized or parallel calcul... more The development of advanced computers with special capabilities for vectorized or parallel calculations demands the development of new calculational methods. The very nature of the Monte Carlo process precludes direct conversion of old (scalar) codes to the new machines. Instead, major changes in global algorithms and careful selection of compatible physics treatments are required. Recent results for Monte Carlo in multigroup shielding applications and in continuous-energy reactor lattice analysis have demonstrated that Monte Carlo methods can be successfully vectorized. The significant effort required for stylized coding and major algorithmic changes is worthwhile, and significant gains in computational efficiency are realized. Speedups of at least twenty to forty times faster than CDC-7600 scalar calculations have been achieved on the CYBER-205 without sacrificing the accuracy of standard Monte Carlo methods. Speedups of this magnitude provide reductions in statistical uncertainties for a given amount of computing time, permit more detailed and realistic problems to be analyzed, and make the Monte Carlo method more accessible to nuclear analysts. Following overviews of the Monte Carlo method for particle transport analysis and of vector computer hardware and software characteristics, both general and specific aspects of the vectorization of Monte Carlo are discussed. Finally, numerical results obtained from vectorized Monte Carlo codes run on the CYBER-205 are presented.
This report was prepared as an account of work sponsored by the United States Government. Neither... more This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product or process disclosed or represents that its use would not infl'inge privately-owned rights.
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Oct 26, 2011
The standard discrete thermal neutron S(a,~) scattering treatment in MCNP5 is compared with a con... more The standard discrete thermal neutron S(a,~) scattering treatment in MCNP5 is compared with a continuous S( a, ~) scattering treatment using a criticality suite of 119 benchmark cases and ENDF IB-VILO nuclear data. In the analysis, six bound isotopes are corisidered: beryllium metal, graphite, hydrogen in water, hydrogen in polyethylene, beryllium in beryllium oxide and oxygen in beryllium oxide. Overall, there are only small changes in the eigenvalue (k eff ) between discrete and continuous treatments. In the comparison of 64 cases that utilize S(a,~) scattering, 62 agreed at the 95% confidence level, and the 2 cases with differences larger than 3 (J agreed exactly when more neutrons were run in the calculations. The results indicate that the changes in eigenvalue between continuous and discrete treatments are random, small, and well within the uncertainty of measured data for reactor criticality experiments.
We describe the initial experience and results from implementing a fission matrix capability into... more We describe the initial experience and results from implementing a fission matrix capability into the MCNP Monte Carlo code. The fission matrix is obtained at essentially no cost during the normal simulation for criticality calculations. It can be used to provide estimates of the fundamental mode fission distribution, the dominance ratio, the eigenvalue spectrum, and higher mode spatial eigenfunctions. It can also be used to accelerate the convergence of the power method iterations and to provide basis functions for higher-order perturbation theory. Past difficulties and limitations of the fission matrix approach are overcome with a new sparse representation of the matrix, permitting much larger and more accurate fission matrix representations. The new fission matrix capabilities provide a significant advance in the state-of-the-art for Monte Carlo criticality calculations.! LA-UR-12-xxxxx Monte Carlo Criticality Calculations" Monte Carlo K-effective Calculation" 1. Start with fission source & eigenvalue guess" 2. Repeat until converged:! • Simulate neutrons, save fission sites for next cycle" • Calculate k-eff, renormalize source" 3. Continue iterating & tally quantities of interest! 4! LA-UR-12-xxxxx
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Papers by William R Martin