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Papers by Phani Motamarri

Research paper thumbnail of Tucker-tensor algorithm for large-scale Kohn-Sham density functional theory calculations

Research paper thumbnail of A Tensor-product approach for large scale electronic structure calculations using Kohn–Sham density functional theory

Quantum-mechanical calculations based on Kohn-Sham density functional theory (DFT) played a signi... more Quantum-mechanical calculations based on Kohn-Sham density functional theory (DFT) played a signifi cant role in accurately predicting various aspects of materials behavior over the past decade. The Kohn-Sham approach to DFT reduces the many-body Schrodinger (eigen value) problem of interacting electrons into an equivalent problem of noninteracting electrons in an effective mean fi eld that is governed by electron-density. Despite the reduced computational complexity of Kohn-Sham DFT, large-scale DFT calculations are still computationally very demanding with the resulting computational complexity scaling cubically with number of atoms in a given materials system. Numerical algorithms with reduced scaling behavior which are robust, computationally effi cient and scalable on parallel computing architectures are always desirable to enable simulations at larger scales and on more complex systems. Following this line of thought, this study explores the use of tensor structured methods for ab-initio numerical solution of Kohn-Sham equations arising in DFT calculations. Earlier studies on tensor-structured methods have been quite successful in the accurate calculation of Hartree and the nonlocal exchange operators arising in the Hartree-Fock equations. A recent investigation of low-rank Tucker-type decomposition of the electron-density of large aluminum clusters (obtained from the fi nite-element discretization of orbital free DFT) shows the exponential decay of approximation error with respect to Tucker rank (number of tensor-basis functions in Tucker type representation). The results also indicate a smaller Tucker rank for the accurate representation of the electron density and is only weakly dependent on the system sizes studied. The promising success of tensor-structured techniques in resolving the electronic structure of material systems has enabled us to take a step further. In this study, we propose a systematic way of computing a globally adapted Tucker-type basis for solving the Kohn-Sham DFT problem by using a separable approximation of the Kohn-Sham Hamiltonian. Further, the resulting Kohn-Sham eigenvalue problem is projected into the aforementioned Tucker basis and is solved for ground-state energy using a self-consistent fi eld iteration. The rank of the resulting Tucker representation and the computational complexity of the calculation are examined on representative benchmark examples involving metallic and insulating systems.

Research paper thumbnail of A numerical analysis of the finite-element discretization of orbital-free density functional theory

In the present work, we study various numerical aspects of higher-order finite-element discretiza... more In the present work, we study various numerical aspects of higher-order finite-element discretizations of the non-linear saddle-point formulation of orbital-free density-functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element discretizations. We next study the convergence properties of higher-order finite-element discretizations of orbital-free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as Coulomb singular potential fields. Our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite-element approximations considered in the present study. We finally investigate the computational efficiency afforded by various higher-order finite-element discretizations, which constitutes the main aspect of the present work, by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use mesh coarse-graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far-field electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite-element basis.

Research paper thumbnail of Subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization

Research paper thumbnail of Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet

Discrete And …, Jan 1, 2006

Research paper thumbnail of Structural topology synthesis with dynamics and nonlinearities using equivalent linear systems

Structural and Multidisciplinary Optimization

Topology optimization for nonlinear and dynamic problems is expensive because of the necessity to... more Topology optimization for nonlinear and dynamic problems is expensive because of the necessity to solve the equations of motion at every optimization iteration in order to evaluate the objective function and constraints. In this work, an iterative methodology is developed using the concept of an equivalent linear system for the topology synthesis of structures undergoing nonlinear and dynamic response, using minimal nonlinear response evaluations. The approach uses equivalent loads obtained from nonlinear dynamic analysis to perform optimization iterations, during the course of which the nonlinear and dynamic system is continuously approximated. In this process, the optimization is decoupled from the nonlinear dynamic analysis. Results are presented for various kinds of nonlinear and dynamic problems showing the effectiveness of the developed approach.

Research paper thumbnail of Higher-order adaptive finite-element methods for Kohn-Sham density functional theory

Arxiv preprint arXiv: …, Jan 1, 2012

Research paper thumbnail of Unified analytical solution for dynamic elastic buckling of beams for various boundary conditions and loading rates

International Journal of Mechanical …, Jan 1, 2012

The problem of dynamic elastic buckling of Euler-Bernoulli beams subjected to axial loads is stud... more The problem of dynamic elastic buckling of Euler-Bernoulli beams subjected to axial loads is studied analytically. The dynamic axial loading is accomplished by a constant displacement rate of one end of the beam with respect to the other. The axial loading rates are considered slow enough to obviate the need to account for axial wave propagation effects. The dynamics of the beam is formulated in the modal domain considering the lowest static buckling mode. The dynamic stability of the beam is investigated as a response problem assuming an initial deformed geometry in terms of an eccentricity favouring the mode shape. The governing partial differential equation is condensed into a unified ordinary differential equation for various boundary conditions using a single dimensionless parameter in terms of beam geometry, material properties, loading rate, and appropriate coefficients corresponding to different boundary conditions. The results obtained for the dynamic response of beams for various values of the dimensionless parameter and initial eccentricity suggest that the solutions can be combined into a single unified analytical expression for the dynamic buckling load. It is shown that the corresponding dynamic response curves can also be collapsed into a single curve using the dimensionless peak load and the associated time parameter. The accuracy of the unified analytical expression for dynamic buckling is verified against exact solution of the ordinary differential equation considering three problems in terms of dimensional quantities for different boundary conditions and loading rates. Further the validity of the single mode dynamic buckling formulation is examined by comparing the results obtained for various boundary conditions with the numerical results from the dynamic response of a large number of degree of freedom finite element model of the beam without any restriction on the deformation mode shape. This unified solution has potential application in optimum design for dynamic collapse of truss type structures subjected to dynamic loads.

Research paper thumbnail of Higher-order adaptive finite-element methods for orbital-free density functional theory

Journal of Computational Physics, Jan 1, 2012

In the present work, we study various numerical aspects of higher-order finite-element discretiza... more In the present work, we study various numerical aspects of higher-order finite-element discretizations of the non-linear saddle-point formulation of orbital-free density-functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element discretizations. We next study the convergence properties of higher-order finite-element discretizations of orbital-free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as Coulomb singular potential fields. Our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite-element approximations considered in the present study. We finally investigate the computational efficiency afforded by various higher-order finite-element discretizations, which constitutes the main aspect of the present work, by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use mesh coarse-graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far-field electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite-element basis.

Research paper thumbnail of An energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

J. Mech. Mater. Structures, Jan 1, 2009

This work deals with the formulation and implementation of an energy-momentum conserving algorith... more This work deals with the formulation and implementation of an energy-momentum conserving algorithm for conducting the nonlinear transient analysis of structures, within the framework of stress-based hybrid elements. Hybrid elements, which are based on a two-field variational formulation, are much less susceptible to locking than conventional displacement-based elements within the static framework. We show that this advantage carries over to the transient case, so that not only are the solutions obtained more accurate, but they are obtained in fewer iterations. We demonstrate the efficacy of the algorithm on a wide range of problems such as ones involving dynamic buckling, complicated three-dimensional motions, et cetera.

Research paper thumbnail of Tucker-tensor algorithm for large-scale Kohn-Sham density functional theory calculations

Research paper thumbnail of A Tensor-product approach for large scale electronic structure calculations using Kohn–Sham density functional theory

Quantum-mechanical calculations based on Kohn-Sham density functional theory (DFT) played a signi... more Quantum-mechanical calculations based on Kohn-Sham density functional theory (DFT) played a signifi cant role in accurately predicting various aspects of materials behavior over the past decade. The Kohn-Sham approach to DFT reduces the many-body Schrodinger (eigen value) problem of interacting electrons into an equivalent problem of noninteracting electrons in an effective mean fi eld that is governed by electron-density. Despite the reduced computational complexity of Kohn-Sham DFT, large-scale DFT calculations are still computationally very demanding with the resulting computational complexity scaling cubically with number of atoms in a given materials system. Numerical algorithms with reduced scaling behavior which are robust, computationally effi cient and scalable on parallel computing architectures are always desirable to enable simulations at larger scales and on more complex systems. Following this line of thought, this study explores the use of tensor structured methods for ab-initio numerical solution of Kohn-Sham equations arising in DFT calculations. Earlier studies on tensor-structured methods have been quite successful in the accurate calculation of Hartree and the nonlocal exchange operators arising in the Hartree-Fock equations. A recent investigation of low-rank Tucker-type decomposition of the electron-density of large aluminum clusters (obtained from the fi nite-element discretization of orbital free DFT) shows the exponential decay of approximation error with respect to Tucker rank (number of tensor-basis functions in Tucker type representation). The results also indicate a smaller Tucker rank for the accurate representation of the electron density and is only weakly dependent on the system sizes studied. The promising success of tensor-structured techniques in resolving the electronic structure of material systems has enabled us to take a step further. In this study, we propose a systematic way of computing a globally adapted Tucker-type basis for solving the Kohn-Sham DFT problem by using a separable approximation of the Kohn-Sham Hamiltonian. Further, the resulting Kohn-Sham eigenvalue problem is projected into the aforementioned Tucker basis and is solved for ground-state energy using a self-consistent fi eld iteration. The rank of the resulting Tucker representation and the computational complexity of the calculation are examined on representative benchmark examples involving metallic and insulating systems.

Research paper thumbnail of A numerical analysis of the finite-element discretization of orbital-free density functional theory

In the present work, we study various numerical aspects of higher-order finite-element discretiza... more In the present work, we study various numerical aspects of higher-order finite-element discretizations of the non-linear saddle-point formulation of orbital-free density-functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element discretizations. We next study the convergence properties of higher-order finite-element discretizations of orbital-free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as Coulomb singular potential fields. Our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite-element approximations considered in the present study. We finally investigate the computational efficiency afforded by various higher-order finite-element discretizations, which constitutes the main aspect of the present work, by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use mesh coarse-graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far-field electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite-element basis.

Research paper thumbnail of Subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization

Research paper thumbnail of Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet

Discrete And …, Jan 1, 2006

Research paper thumbnail of Structural topology synthesis with dynamics and nonlinearities using equivalent linear systems

Structural and Multidisciplinary Optimization

Topology optimization for nonlinear and dynamic problems is expensive because of the necessity to... more Topology optimization for nonlinear and dynamic problems is expensive because of the necessity to solve the equations of motion at every optimization iteration in order to evaluate the objective function and constraints. In this work, an iterative methodology is developed using the concept of an equivalent linear system for the topology synthesis of structures undergoing nonlinear and dynamic response, using minimal nonlinear response evaluations. The approach uses equivalent loads obtained from nonlinear dynamic analysis to perform optimization iterations, during the course of which the nonlinear and dynamic system is continuously approximated. In this process, the optimization is decoupled from the nonlinear dynamic analysis. Results are presented for various kinds of nonlinear and dynamic problems showing the effectiveness of the developed approach.

Research paper thumbnail of Higher-order adaptive finite-element methods for Kohn-Sham density functional theory

Arxiv preprint arXiv: …, Jan 1, 2012

Research paper thumbnail of Unified analytical solution for dynamic elastic buckling of beams for various boundary conditions and loading rates

International Journal of Mechanical …, Jan 1, 2012

The problem of dynamic elastic buckling of Euler-Bernoulli beams subjected to axial loads is stud... more The problem of dynamic elastic buckling of Euler-Bernoulli beams subjected to axial loads is studied analytically. The dynamic axial loading is accomplished by a constant displacement rate of one end of the beam with respect to the other. The axial loading rates are considered slow enough to obviate the need to account for axial wave propagation effects. The dynamics of the beam is formulated in the modal domain considering the lowest static buckling mode. The dynamic stability of the beam is investigated as a response problem assuming an initial deformed geometry in terms of an eccentricity favouring the mode shape. The governing partial differential equation is condensed into a unified ordinary differential equation for various boundary conditions using a single dimensionless parameter in terms of beam geometry, material properties, loading rate, and appropriate coefficients corresponding to different boundary conditions. The results obtained for the dynamic response of beams for various values of the dimensionless parameter and initial eccentricity suggest that the solutions can be combined into a single unified analytical expression for the dynamic buckling load. It is shown that the corresponding dynamic response curves can also be collapsed into a single curve using the dimensionless peak load and the associated time parameter. The accuracy of the unified analytical expression for dynamic buckling is verified against exact solution of the ordinary differential equation considering three problems in terms of dimensional quantities for different boundary conditions and loading rates. Further the validity of the single mode dynamic buckling formulation is examined by comparing the results obtained for various boundary conditions with the numerical results from the dynamic response of a large number of degree of freedom finite element model of the beam without any restriction on the deformation mode shape. This unified solution has potential application in optimum design for dynamic collapse of truss type structures subjected to dynamic loads.

Research paper thumbnail of Higher-order adaptive finite-element methods for orbital-free density functional theory

Journal of Computational Physics, Jan 1, 2012

In the present work, we study various numerical aspects of higher-order finite-element discretiza... more In the present work, we study various numerical aspects of higher-order finite-element discretizations of the non-linear saddle-point formulation of orbital-free density-functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element discretizations. We next study the convergence properties of higher-order finite-element discretizations of orbital-free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as Coulomb singular potential fields. Our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite-element approximations considered in the present study. We finally investigate the computational efficiency afforded by various higher-order finite-element discretizations, which constitutes the main aspect of the present work, by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use mesh coarse-graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far-field electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite-element basis.

Research paper thumbnail of An energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

J. Mech. Mater. Structures, Jan 1, 2009

This work deals with the formulation and implementation of an energy-momentum conserving algorith... more This work deals with the formulation and implementation of an energy-momentum conserving algorithm for conducting the nonlinear transient analysis of structures, within the framework of stress-based hybrid elements. Hybrid elements, which are based on a two-field variational formulation, are much less susceptible to locking than conventional displacement-based elements within the static framework. We show that this advantage carries over to the transient case, so that not only are the solutions obtained more accurate, but they are obtained in fewer iterations. We demonstrate the efficacy of the algorithm on a wide range of problems such as ones involving dynamic buckling, complicated three-dimensional motions, et cetera.