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Papers by Thomas Blesgen

For the validity of the weak maximum principle for classical solutions of elliptic partial differ... more For the validity of the weak maximum principle for classical solutions of elliptic partial differential equations it is sufficient that the coefficient matrix a ij (x) is non-negative. In this note we consider maximum principles for weak solutions of elliptic partial differential equations in divergence form with bounded coefficients a ij. We demonstrate that the assumption that the coefficient matrix a ij (x) is positive almost everywhere is essential and cannot be weakened. To this end we give a counter example originating from geometrically linear elasticity. Note After submission of this paper we learned that there are much simpler examples which demonstrate that the positivity of the coefficient matrix a ij (x) is essential and cannot be weakened. In two space dimensions, let the coefficient matrix be given by a ij (x 1 , x 2) = x 2 2

Non-periodic finite-element formulation of Kohn–Sham density functional theory

arXiv (Cornell University), Feb 23, 2012

We present an extension of the Allen-Cahn/Cahn-Hilliard system which incorporates a geometrically... more We present an extension of the Allen-Cahn/Cahn-Hilliard system which incorporates a geometrically linear ansatz for the elastic energy of the precipitates. The model contains both the elastic Allen-Cahn system and the elastic Cahn-Hilliard system as special cases and accounts for the microstructures on the microscopic scale. We prove the existence of weak solutions to the new model for a general class of energy functionals. We then give several examples of functionals that belong to this class. This includes the energy of geometrically linear elastic materials for D < 3. Moreover we show this for D = 3 in the setting of scalar-valued deformations, which corresponds to the case of anti-plane shear. All this is based on explicit formulas for relaxed energy functionals newly derived in this article for D = 1 and D = 3. In these cases we can also prove uniqueness of the weak solutions.

arXiv (Cornell University), Dec 10, 2011

Using chains of bistable springs, a model is derived to investigate the plastic behavior of carbo... more Using chains of bistable springs, a model is derived to investigate the plastic behavior of carbon nanotube arrays with damage. We study the preconditioning effect due to the loading history by computing analytically the stress-strain pattern corresponding to a fatigue-type damage of the structure. We identify the convergence of the discrete response to the limiting case of infinitely many springs, both analytically in the framework of Gamma-convergence, as well as numerically.

Physical review, Mar 1, 2016

In this work, we propose a systematic way of computing a low-rank globally-adapted localized Tuck... more In this work, we propose a systematic way of computing a low-rank globally-adapted localized Tucker-tensor basis for solving the Kohn-Sham DFT problem. In every iteration of the self-consistent field procedure of the Kohn-Sham DFT problem, we construct an additive separable approximation of the Kohn-Sham Hamiltonian. The Tucker-tensor basis is chosen such as to span the tensor product of the one-dimensional eigenspaces corresponding to each of the spatially separable Hamiltonians, and the localized Tucker-tensor basis is constructed from localized representations of these one-dimensional eigenspaces. This Tucker-tensor basis forms a complete basis, and is naturally adapted to the Kohn-Sham Hamiltonian. Further, the locality of this basis in real-space allows us to exploit reduced-order scaling algorithms for the solution of the discrete Kohn-Sham eigenvalue problem. In particular, we use Chebyshev filtering to compute the eigenspace of the Kohn-Sham Hamiltonian, and evaluate non-orthogonal localized wavefunctions spanning the Chebyshev filtered space, all represented in the Tucker-tensor basis. We thereby compute the electron-density and other quantities of interest, using a Fermi-operator expansion of the Hamiltonian projected onto the subspace spanned by the non-orthogonal localized wavefunctions. Numerical results on benchmark examples involving pseudopotential calculations suggest an exponential convergence of the groundstate energy with the Tucker rank. Interestingly, the rank of the Tucker-tensor basis required to obtain chemical accuracy is found to be only weakly dependent on the system size, which results in close to linear-scaling complexity for Kohn-Sham DFT calculations for both insulating and metallic systems. A comparative study has revealed significant computational efficiencies afforded by the proposed Tucker-tensor approach in comparison to a plane-wave basis.

Springer eBooks, Oct 14, 2006

We discuss two different approaches related to Γ-limits of free energy functionals. The first giv... more We discuss two different approaches related to Γ-limits of free energy functionals. The first gives an example of how symmetry breaking may occur on the atomistic level, the second aims at deriving a general analytic theory for elasticity on the lattice scale that does not depend on an explicitly chosen reference system.

Journal of The Mechanics and Physics of Solids, Feb 1, 2010

We present a real-space, non-periodic, finite-element formulation for Kohn-Sham Density Functiona... more We present a real-space, non-periodic, finite-element formulation for Kohn-Sham Density Functional Theory (KS-DFT). We transform the original variational problem into a local saddle-point problem, and show its well-posedness by proving the existence of minimizers. Further, we prove the convergence of finite-element approximations including numerical quadratures. Based on domain decomposition, we develop a parallel finite-element implementation of this formulation capable of performing both all-electron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature. We also evaluate the numerical performance of the implementation with regard to its scalability and convergence rates. We view this work as a step towards developing a method that can accurately study defects like vacancies, dislocations and crack tips using Density Functional Theory (DFT) at reasonable computational cost by retaining electronic resolution where it is necessary and seamlessly coarse-graining far away.

arXiv (Cornell University), Feb 18, 2023

The zero and first order Gamma-limit of vanishing internal length scale are studied for the mecha... more The zero and first order Gamma-limit of vanishing internal length scale are studied for the mechanical energy of a shear problem in geometrically nonlinear Cosserat elasticity. The convergence of the minimizers is shown and the limit functionals are characterized.

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.

Journal of The Mechanics and Physics of Solids, 2011

We present some additions to Section 5 of the article 'Multiscale mass-spring models of carbon na... more We present some additions to Section 5 of the article 'Multiscale mass-spring models of carbon nanotube foams', to be published online alongside its electronic version.

Mathematics and Mechanics of Solids

Deformation microstructure is studied for a 1 D-shear problem in geometrically nonlinear Cosserat... more Deformation microstructure is studied for a 1 D-shear problem in geometrically nonlinear Cosserat elasticity. Microstructure solutions are described analytically and numerically for zero characteristic length scale.

arXiv (Cornell University), Jun 16, 2022

Deformation microstructure is studied for a 1D-shear problem in geometrically nonlinear Cosserat ... more Deformation microstructure is studied for a 1D-shear problem in geometrically nonlinear Cosserat elasticity. Microstructure solutions are described analytically and numerically for zero characteristic length scale.

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.

Analysis, Modeling and Simulation of Multiscale Problems

We discuss two different approaches related to Γ-limits of free energy functionals. The first giv... more We discuss two different approaches related to Γ-limits of free energy functionals. The first gives an example of how symmetry breaking may occur on the atomistic level, the second aims at deriving a general analytic theory for elasticity on the lattice scale that does not depend on an explicitly chosen reference system.

A model describing phase transitions in crystals coupled with diusion and linear elasticity under... more A model describing phase transitions in crystals coupled with diusion and linear elasticity under isothermal conditions is studied numerically in two space dimensions. For the minimisation of the free energy w.r.t. the phase parameter an algorithm based on the level set approach is presented to- gether with an extension based on the isoperimet- ric inequality that allows to find a global minimum in most cases.

Decay estimates for the electronic den-sity in equations of Schrödinger type as the Helmholtz pro... more Decay estimates for the electronic den-sity in equations of Schrödinger type as the Helmholtz problem are derived. For radially-symmetric solu-tions, these decay estimates are improved and made quantitative. Decay results of the elastic field are also recalled. The analytic results are applied to imple-ment a coarsening strategy to numerically solve the Kohn-Sham problem of quantum mechanics for multi-millions of atoms. In the framework of Γ-convergence, minimizers of the Rayleigh quotient in the weighted L 2 -topology are studied and monotonicity results for the perturbed and the unperturbed eigenvalue prob-lem are established. In one-dimensional case-studies typical properties of the solutions are investigated and non-exponential decay in 1D is demonstrated.

Multiscale Modeling & Simulation, 2013

Based on a one-dimensional discrete system of bistable springs, a mechanical model is introduced ... more Based on a one-dimensional discrete system of bistable springs, a mechanical model is introduced to describe plasticity and damage in carbon nanotube (CNT) arrays. The energetics of the mechanical system are investigated analytically, the stress-strain law is derived, and the mechanical dissipation is computed, both for the discrete case as well as for the continuum limit. An information-passing approach is developed that permits the investigation of macroscopic portions of the material. As an application, the simulation of a cyclic compression experiment on real CNT foam is performed, considering both the material response during the primary loading path from the virgin state and the damaged response after preconditioning.

Meccanica, 2019

This article deals with the mathematical derivation and the validation over benchmark examples of... more This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler-Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

ArXiv, 2019

This article deals with the mathematical derivation and the validation over benchmark examples of... more This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler-Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

Continuum Mechanics and Thermodynamics, 2019

This work reconsiders the Becker-Döring model for nucleation under isothermal conditions in the p... more This work reconsiders the Becker-Döring model for nucleation under isothermal conditions in the presence of phase transitions. Based on thermodynamic principles a modified model is derived where the condensation and evaporation rates may depend on the phase parameter. The existence and uniqueness of weak solutions to the proposed model are shown and the corresponding equilibrium states are characterized in terms of response functions and constitutive variables.

For the validity of the weak maximum principle for classical solutions of elliptic partial differ... more For the validity of the weak maximum principle for classical solutions of elliptic partial differential equations it is sufficient that the coefficient matrix a ij (x) is non-negative. In this note we consider maximum principles for weak solutions of elliptic partial differential equations in divergence form with bounded coefficients a ij. We demonstrate that the assumption that the coefficient matrix a ij (x) is positive almost everywhere is essential and cannot be weakened. To this end we give a counter example originating from geometrically linear elasticity. Note After submission of this paper we learned that there are much simpler examples which demonstrate that the positivity of the coefficient matrix a ij (x) is essential and cannot be weakened. In two space dimensions, let the coefficient matrix be given by a ij (x 1 , x 2) = x 2 2

Non-periodic finite-element formulation of Kohn–Sham density functional theory

arXiv (Cornell University), Feb 23, 2012

We present an extension of the Allen-Cahn/Cahn-Hilliard system which incorporates a geometrically... more We present an extension of the Allen-Cahn/Cahn-Hilliard system which incorporates a geometrically linear ansatz for the elastic energy of the precipitates. The model contains both the elastic Allen-Cahn system and the elastic Cahn-Hilliard system as special cases and accounts for the microstructures on the microscopic scale. We prove the existence of weak solutions to the new model for a general class of energy functionals. We then give several examples of functionals that belong to this class. This includes the energy of geometrically linear elastic materials for D < 3. Moreover we show this for D = 3 in the setting of scalar-valued deformations, which corresponds to the case of anti-plane shear. All this is based on explicit formulas for relaxed energy functionals newly derived in this article for D = 1 and D = 3. In these cases we can also prove uniqueness of the weak solutions.

arXiv (Cornell University), Dec 10, 2011

Using chains of bistable springs, a model is derived to investigate the plastic behavior of carbo... more Using chains of bistable springs, a model is derived to investigate the plastic behavior of carbon nanotube arrays with damage. We study the preconditioning effect due to the loading history by computing analytically the stress-strain pattern corresponding to a fatigue-type damage of the structure. We identify the convergence of the discrete response to the limiting case of infinitely many springs, both analytically in the framework of Gamma-convergence, as well as numerically.

Physical review, Mar 1, 2016

In this work, we propose a systematic way of computing a low-rank globally-adapted localized Tuck... more In this work, we propose a systematic way of computing a low-rank globally-adapted localized Tucker-tensor basis for solving the Kohn-Sham DFT problem. In every iteration of the self-consistent field procedure of the Kohn-Sham DFT problem, we construct an additive separable approximation of the Kohn-Sham Hamiltonian. The Tucker-tensor basis is chosen such as to span the tensor product of the one-dimensional eigenspaces corresponding to each of the spatially separable Hamiltonians, and the localized Tucker-tensor basis is constructed from localized representations of these one-dimensional eigenspaces. This Tucker-tensor basis forms a complete basis, and is naturally adapted to the Kohn-Sham Hamiltonian. Further, the locality of this basis in real-space allows us to exploit reduced-order scaling algorithms for the solution of the discrete Kohn-Sham eigenvalue problem. In particular, we use Chebyshev filtering to compute the eigenspace of the Kohn-Sham Hamiltonian, and evaluate non-orthogonal localized wavefunctions spanning the Chebyshev filtered space, all represented in the Tucker-tensor basis. We thereby compute the electron-density and other quantities of interest, using a Fermi-operator expansion of the Hamiltonian projected onto the subspace spanned by the non-orthogonal localized wavefunctions. Numerical results on benchmark examples involving pseudopotential calculations suggest an exponential convergence of the groundstate energy with the Tucker rank. Interestingly, the rank of the Tucker-tensor basis required to obtain chemical accuracy is found to be only weakly dependent on the system size, which results in close to linear-scaling complexity for Kohn-Sham DFT calculations for both insulating and metallic systems. A comparative study has revealed significant computational efficiencies afforded by the proposed Tucker-tensor approach in comparison to a plane-wave basis.

Springer eBooks, Oct 14, 2006

We discuss two different approaches related to Γ-limits of free energy functionals. The first giv... more We discuss two different approaches related to Γ-limits of free energy functionals. The first gives an example of how symmetry breaking may occur on the atomistic level, the second aims at deriving a general analytic theory for elasticity on the lattice scale that does not depend on an explicitly chosen reference system.

Journal of The Mechanics and Physics of Solids, Feb 1, 2010

We present a real-space, non-periodic, finite-element formulation for Kohn-Sham Density Functiona... more We present a real-space, non-periodic, finite-element formulation for Kohn-Sham Density Functional Theory (KS-DFT). We transform the original variational problem into a local saddle-point problem, and show its well-posedness by proving the existence of minimizers. Further, we prove the convergence of finite-element approximations including numerical quadratures. Based on domain decomposition, we develop a parallel finite-element implementation of this formulation capable of performing both all-electron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature. We also evaluate the numerical performance of the implementation with regard to its scalability and convergence rates. We view this work as a step towards developing a method that can accurately study defects like vacancies, dislocations and crack tips using Density Functional Theory (DFT) at reasonable computational cost by retaining electronic resolution where it is necessary and seamlessly coarse-graining far away.

arXiv (Cornell University), Feb 18, 2023

The zero and first order Gamma-limit of vanishing internal length scale are studied for the mecha... more The zero and first order Gamma-limit of vanishing internal length scale are studied for the mechanical energy of a shear problem in geometrically nonlinear Cosserat elasticity. The convergence of the minimizers is shown and the limit functionals are characterized.

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.

Journal of The Mechanics and Physics of Solids, 2011

We present some additions to Section 5 of the article 'Multiscale mass-spring models of carbon na... more We present some additions to Section 5 of the article 'Multiscale mass-spring models of carbon nanotube foams', to be published online alongside its electronic version.

Mathematics and Mechanics of Solids

Deformation microstructure is studied for a 1 D-shear problem in geometrically nonlinear Cosserat... more Deformation microstructure is studied for a 1 D-shear problem in geometrically nonlinear Cosserat elasticity. Microstructure solutions are described analytically and numerically for zero characteristic length scale.

arXiv (Cornell University), Jun 16, 2022

Deformation microstructure is studied for a 1D-shear problem in geometrically nonlinear Cosserat ... more Deformation microstructure is studied for a 1D-shear problem in geometrically nonlinear Cosserat elasticity. Microstructure solutions are described analytically and numerically for zero characteristic length scale.

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.

Analysis, Modeling and Simulation of Multiscale Problems

We discuss two different approaches related to Γ-limits of free energy functionals. The first giv... more We discuss two different approaches related to Γ-limits of free energy functionals. The first gives an example of how symmetry breaking may occur on the atomistic level, the second aims at deriving a general analytic theory for elasticity on the lattice scale that does not depend on an explicitly chosen reference system.

A model describing phase transitions in crystals coupled with diusion and linear elasticity under... more A model describing phase transitions in crystals coupled with diusion and linear elasticity under isothermal conditions is studied numerically in two space dimensions. For the minimisation of the free energy w.r.t. the phase parameter an algorithm based on the level set approach is presented to- gether with an extension based on the isoperimet- ric inequality that allows to find a global minimum in most cases.

Decay estimates for the electronic den-sity in equations of Schrödinger type as the Helmholtz pro... more Decay estimates for the electronic den-sity in equations of Schrödinger type as the Helmholtz problem are derived. For radially-symmetric solu-tions, these decay estimates are improved and made quantitative. Decay results of the elastic field are also recalled. The analytic results are applied to imple-ment a coarsening strategy to numerically solve the Kohn-Sham problem of quantum mechanics for multi-millions of atoms. In the framework of Γ-convergence, minimizers of the Rayleigh quotient in the weighted L 2 -topology are studied and monotonicity results for the perturbed and the unperturbed eigenvalue prob-lem are established. In one-dimensional case-studies typical properties of the solutions are investigated and non-exponential decay in 1D is demonstrated.

Multiscale Modeling & Simulation, 2013

Based on a one-dimensional discrete system of bistable springs, a mechanical model is introduced ... more Based on a one-dimensional discrete system of bistable springs, a mechanical model is introduced to describe plasticity and damage in carbon nanotube (CNT) arrays. The energetics of the mechanical system are investigated analytically, the stress-strain law is derived, and the mechanical dissipation is computed, both for the discrete case as well as for the continuum limit. An information-passing approach is developed that permits the investigation of macroscopic portions of the material. As an application, the simulation of a cyclic compression experiment on real CNT foam is performed, considering both the material response during the primary loading path from the virgin state and the damaged response after preconditioning.

Meccanica, 2019

This article deals with the mathematical derivation and the validation over benchmark examples of... more This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler-Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

ArXiv, 2019

This article deals with the mathematical derivation and the validation over benchmark examples of... more This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler-Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

Continuum Mechanics and Thermodynamics, 2019

This work reconsiders the Becker-Döring model for nucleation under isothermal conditions in the p... more This work reconsiders the Becker-Döring model for nucleation under isothermal conditions in the presence of phase transitions. Based on thermodynamic principles a modified model is derived where the condensation and evaporation rates may depend on the phase parameter. The existence and uniqueness of weak solutions to the proposed model are shown and the corresponding equilibrium states are characterized in terms of response functions and constitutive variables.