Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous f... more Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses O͑N log N͒ operations and O͑N͒ memory, where N is the number of grid points. This system-size scaling is significant because of the very large N required for three-dimensional systems. The algorithm uses fast Fourier transforms ͑FFTs͒ to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard ͑iterative substitution͒ iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT, we use fundamental measure theory. For the electrostatic DFT, we present two algorithms. One is for the "bulk-fluid" functional of Rosenfeld ͓Y. Rosenfeld, J. Chem. Phys. 98, 8126 ͑1993͔͒ that uses O͑N log N͒ operations. The other is for the "reference fluid density" ͑RFD͒ functional ͓D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 ͑2002͔͒. This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires O͑N 2 ͒ operations.
A new representation of computational meshes is proposed in terms of a covering relation defined ... more A new representation of computational meshes is proposed in terms of a covering relation defined by discrete topological objects we call sieves. Fields over a mesh are handled locally by using the notion of refinement, dual to covering, and are later reassembled. In this approach fields are modeled by sections of a fiber bundle over a sieve. This approach cleanly separates the topology of the mesh from its geometry and other value-storage mechanisms. With these abstractions, finite element calculations are expressed using algorithms that are independent of mesh dimension, global topology, element shapes, and the finite element itself. Extensions and other applications are discussed.
We consider methods for systematic construction of algorithms for a class of time-dependent PDEs ... more We consider methods for systematic construction of algorithms for a class of time-dependent PDEs with Hamiltonian structure. These systems possess phase space geometry and constants of the motion that need to be preserved by the integration algorithm to reflect the qualitative features of the system. We exploit the structure of Hamiltonian systems, in particular their variational formulation based on a Lagrangian, and the dual covariant formulation, to expose the geometric features of the system that have natural analogs when discretized. We emphasize the local space-time approach to the constructions, making them amenable to parallelization and preconditioning using domain decomposition methods, and enabling treatment of complex spatial geometries and boundary conditions. These methods are applied to the two representative problems: the Nonlinear Schrodinger equation (NLS) and the Heisenberg magnet model (HM), both of which possess highly nontrivial geometric structures. We treat t...
We have developed a new programming framework, called Sieve, to support parallel numerical PDE al... more We have developed a new programming framework, called Sieve, to support parallel numerical PDE algorithms operating over distributed meshes. We have also developed a reference implementation of Sieve in C++ as a library of generic algorithms operating on distributed containers conforming to the Sieve interface. Sieve makes instances of the incidence relation, or \emph{arrows}, the conceptual first-class objects represented in
We have developed a new programming framework, called Sieve, to support parallel numerical PDE 1 ... more We have developed a new programming framework, called Sieve, to support parallel numerical PDE 1 algorithms operating over distributed meshes. We have also developed a reference implementation of Sieve in C++ as a library of generic algorithms operating on distributed containers conforming to the Sieve interface. Sieve makes instances of the incidence relation, or arrows, the conceptual first-class objects represented in the containers. Further, generic algorithms acting on this arrow container are systematically used to provide natural geometric operations on the topology and also, through duality, on the data. Finally, coverings and duality are used to encode not only individual meshes, but all types of hierarchies underlying PDE data structures, including multigrid and mesh partitions.
We perform Monte-Carlo-type simulation studies of self-organization of microtubules interacting w... more We perform Monte-Carlo-type simulation studies of self-organization of microtubules interacting with molecular motors. We model microtubules as stiff polar rods of equal length exhibiting anisotropic diffusion in the plane. The molecular motors are implicitly introduced by specifying certain probabilistic collision rules resulting in realignment of the rods. This approximation of the complicated microtubule-motor interaction by simple a instant collision allows us to by-pass the "computational bottlenecks" associated with the diffusion and dynamics of the motors and the reorientation of the microtubules. Consequently, we are able to perform simulations of large ensembles of microtubules and motors on a very large time scale. This simple model reproduces all important phenomenology observed in in vitro experiments: formation of vortices for low motor density and ray-like asters and bundles for higher motor density.
This paper summarizes the results of numerical simulations of the interaction of a pair of biofil... more This paper summarizes the results of numerical simulations of the interaction of a pair of biofilaments mediated by a molecular motor. The filaments are modeled as flexible rods, and the results are applicable to microtubules, which are relatively stiff, as well as to much softer filaments, such as actin. The results provide insight into the effects of flexibility on cytoskeleton formation and the rheology of semiflexible filament networks. The simulations are based on a nonlinear elasticity equation. The results show that flexibility enhances the tendency of filaments to align. The enhancement in turn favors the formation of large-scale structures in multifilament systems. Simulations for soft filaments show that the action of the motor can result in the formation of multiple loops of the filaments as a result of buckling, which can affect the structure of a cross-linked network and thereby its rheology. The estimate for the minimal buckling length as a function of the motor speed, the viscosity of the solvent, and the bending stiffness of the filament is derived analytically.
Suspensions of self-propelled particles are studied in the framework of twodimensional (2D) Stoke... more Suspensions of self-propelled particles are studied in the framework of twodimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions.
We have recently developed two methods to calculate the magnetic normal modes of a magnetic nano-... more We have recently developed two methods to calculate the magnetic normal modes of a magnetic nano-particle. One of the methods is based on a conventional micromagnetic approach in which the time evolution of the magnetization of each cell is monitored. After filtering in frequency domain, the magnetic normal modes can be reconstructed. The second method is based on solving the same micromagneitc system in a dynamical matrix formulation. The results of the two methods, applied to a rectangular parallelepiped of Fe, will be presented and compared. r
In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (... more In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (HM) model which are then used as the basis for geometric discretizations of HM. Despite a topological obstruction to the existence of a global Lagrangian density, a local variational formulation allows one to derive local conservation laws using a version of Nöther's theorem from the formal variational calculus of Gelfand-Dikii. Using the local Lagrangian form we extend the method of Marsden, Patrick and Schkoller to derive local multisymplectic discretizations directly from the variational principle. We employ a version of the finite element method to discretize the space of sections of the trivial magnetic spin bundle N = M × S 2 over an appropriate space-time M . Since sections do not form a vector space, the usual FEM bases can be used only locally with coordinate transformations intervening on element boundaries, and conservation properties are guaranteed only within an element. We discuss possible ways of circumventing this problem, including the use of a local version of the method of characteristics, non-polynomial FEM bases and Lie-group discretization methods.
Eective viscosity (EV) of suspensions of puller-like microswimmers (pullers), such as the Chlamyd... more Eective viscosity (EV) of suspensions of puller-like microswimmers (pullers), such as the Chlamydamonas algae, is dicult to measure or to simulate for all swimmer concentrations. Although there are good reasons to expect that the EV of pullers is similar to that of passive suspensions, analytic determination of the passive EV for all concentrations remains unsatisfactory. At the same time, the EV of bacterial suspensions is closely linked to collective motion in these systems and is biologically signicant. We develop an approach to determining analytic EV estimates at all concentrations for suspensions of pullers as well as for passive suspensions.The proposed methods are based on the ideas of renormalization group theory (RG) and construct the EV formula based on the known asymptotics for small concentrations and near the critical point (i.e., approaching dense packing). For passive suspensions the method is veried by comparison against known theoretical results. We nd that the method performs much better than an earlier RG-based technique. For pullers the validation is done by comparing to experiments conducted on Chlamydamonas suspensions.
Recently an interesting new class of PDE integrators, multisymplectic schemes, has been introduce... more Recently an interesting new class of PDE integrators, multisymplectic schemes, has been introduced for solving systems possessing a certain multisymplectic structure. Some of the characteristic features of the method are its local nature (independent of boundary conditions) and an equal treatment of spatial and temporal variables. The nonlinear Schrödinger equation (NLS) has a multisymplectic formulation, and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the NLS. In the numerical experiments, the multisymplectic concatenated midpoint scheme (a centered cell discretization) is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff. On the other hand, an integrable Hamiltonian semi-discretization of NLS from Ablowitz and Ladik (AL) possesses a full set of global conservation laws and a noncanonical symplectic structure. We generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the AL type. Another approach examined in the paper is the introduction of transformations to reduce the AL system to either (1) separable form or (2) canonical form and then apply standard schemes in the new coordinates. All of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions. The performance of the schemes as well as interrelations among various geometric features are discussed.
We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbel... more We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by "pushing" or "pulling" both in three-and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria.
on three factors; the dipole moment of the coordinating ligands [1], the number of coordinating l... more on three factors; the dipole moment of the coordinating ligands [1], the number of coordinating ligands [2] and the cavity size of the coordination site [3]. By using free energy methods in molecular dynamics simulations to study model systems as well as a range of biological molecules including channels, transporters and enzymes, we are able to determine the contribution of each of these factors to the overall ion selectivity of the molecule. Varying contributions of each give rise to the richness in ion selectivity we see. By mapping out the importance of these factors in various cases, an estimation of the degree of selectivity of an ion selective biological molecule can be made given its structure. These results also assist in predicting the nature of a binding site of an unknown structure and have the potential to aid in the design of novel synthetic ion selective molecules.
A new representation of computational meshes is proposed in terms of a covering relation defined ... more A new representation of computational meshes is proposed in terms of a covering relation defined by discrete topological objects we call sieves. Fields over a mesh are handled locally by using the notion of refinement, dual to covering, and are later reassembled. In this approach fields are modeled by sections of a fiber bundle over a sieve. This approach cleanly separates the topology of the mesh from its geometry and other value-storage mechanisms. With these abstractions, finite element calculations are expressed using algorithms that are independent of mesh dimension, global topology, element shapes, and the finite element itself. Extensions and other applications are discussed. Despite what the name might suggest no extra relations between the elements of the chain are presumed. From the point of view of combinatorial topology we are dealing with chains mod 2 or chains with coefficients in the two-point field F 2 . However, we shall not use the algebraic structure or homology in this publication. 2 The obvious analogy with the categorical notions of product and coproduct will be clarified in our forthcoming paper .
Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous f... more Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses O͑N log N͒ operations and O͑N͒ memory, where N is the number of grid points. This system-size scaling is significant because of the very large N required for three-dimensional systems. The algorithm uses fast Fourier transforms ͑FFTs͒ to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard ͑iterative substitution͒ iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT, we use fundamental measure theory. For the electrostatic DFT, we present two algorithms. One is for the "bulk-fluid" functional of Rosenfeld ͓Y. Rosenfeld, J. Chem. Phys. 98, 8126 ͑1993͔͒ that uses O͑N log N͒ operations. The other is for the "reference fluid density" ͑RFD͒ functional ͓D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 ͑2002͔͒. This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires O͑N 2 ͒ operations.
Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous f... more Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses O͑N log N͒ operations and O͑N͒ memory, where N is the number of grid points. This system-size scaling is significant because of the very large N required for three-dimensional systems. The algorithm uses fast Fourier transforms ͑FFTs͒ to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard ͑iterative substitution͒ iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT, we use fundamental measure theory. For the electrostatic DFT, we present two algorithms. One is for the "bulk-fluid" functional of Rosenfeld ͓Y. Rosenfeld, J. Chem. Phys. 98, 8126 ͑1993͔͒ that uses O͑N log N͒ operations. The other is for the "reference fluid density" ͑RFD͒ functional ͓D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 ͑2002͔͒. This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires O͑N 2 ͒ operations.
A new representation of computational meshes is proposed in terms of a covering relation defined ... more A new representation of computational meshes is proposed in terms of a covering relation defined by discrete topological objects we call sieves. Fields over a mesh are handled locally by using the notion of refinement, dual to covering, and are later reassembled. In this approach fields are modeled by sections of a fiber bundle over a sieve. This approach cleanly separates the topology of the mesh from its geometry and other value-storage mechanisms. With these abstractions, finite element calculations are expressed using algorithms that are independent of mesh dimension, global topology, element shapes, and the finite element itself. Extensions and other applications are discussed.
We consider methods for systematic construction of algorithms for a class of time-dependent PDEs ... more We consider methods for systematic construction of algorithms for a class of time-dependent PDEs with Hamiltonian structure. These systems possess phase space geometry and constants of the motion that need to be preserved by the integration algorithm to reflect the qualitative features of the system. We exploit the structure of Hamiltonian systems, in particular their variational formulation based on a Lagrangian, and the dual covariant formulation, to expose the geometric features of the system that have natural analogs when discretized. We emphasize the local space-time approach to the constructions, making them amenable to parallelization and preconditioning using domain decomposition methods, and enabling treatment of complex spatial geometries and boundary conditions. These methods are applied to the two representative problems: the Nonlinear Schrodinger equation (NLS) and the Heisenberg magnet model (HM), both of which possess highly nontrivial geometric structures. We treat t...
We have developed a new programming framework, called Sieve, to support parallel numerical PDE al... more We have developed a new programming framework, called Sieve, to support parallel numerical PDE algorithms operating over distributed meshes. We have also developed a reference implementation of Sieve in C++ as a library of generic algorithms operating on distributed containers conforming to the Sieve interface. Sieve makes instances of the incidence relation, or \emph{arrows}, the conceptual first-class objects represented in
We have developed a new programming framework, called Sieve, to support parallel numerical PDE 1 ... more We have developed a new programming framework, called Sieve, to support parallel numerical PDE 1 algorithms operating over distributed meshes. We have also developed a reference implementation of Sieve in C++ as a library of generic algorithms operating on distributed containers conforming to the Sieve interface. Sieve makes instances of the incidence relation, or arrows, the conceptual first-class objects represented in the containers. Further, generic algorithms acting on this arrow container are systematically used to provide natural geometric operations on the topology and also, through duality, on the data. Finally, coverings and duality are used to encode not only individual meshes, but all types of hierarchies underlying PDE data structures, including multigrid and mesh partitions.
We perform Monte-Carlo-type simulation studies of self-organization of microtubules interacting w... more We perform Monte-Carlo-type simulation studies of self-organization of microtubules interacting with molecular motors. We model microtubules as stiff polar rods of equal length exhibiting anisotropic diffusion in the plane. The molecular motors are implicitly introduced by specifying certain probabilistic collision rules resulting in realignment of the rods. This approximation of the complicated microtubule-motor interaction by simple a instant collision allows us to by-pass the "computational bottlenecks" associated with the diffusion and dynamics of the motors and the reorientation of the microtubules. Consequently, we are able to perform simulations of large ensembles of microtubules and motors on a very large time scale. This simple model reproduces all important phenomenology observed in in vitro experiments: formation of vortices for low motor density and ray-like asters and bundles for higher motor density.
This paper summarizes the results of numerical simulations of the interaction of a pair of biofil... more This paper summarizes the results of numerical simulations of the interaction of a pair of biofilaments mediated by a molecular motor. The filaments are modeled as flexible rods, and the results are applicable to microtubules, which are relatively stiff, as well as to much softer filaments, such as actin. The results provide insight into the effects of flexibility on cytoskeleton formation and the rheology of semiflexible filament networks. The simulations are based on a nonlinear elasticity equation. The results show that flexibility enhances the tendency of filaments to align. The enhancement in turn favors the formation of large-scale structures in multifilament systems. Simulations for soft filaments show that the action of the motor can result in the formation of multiple loops of the filaments as a result of buckling, which can affect the structure of a cross-linked network and thereby its rheology. The estimate for the minimal buckling length as a function of the motor speed, the viscosity of the solvent, and the bending stiffness of the filament is derived analytically.
Suspensions of self-propelled particles are studied in the framework of twodimensional (2D) Stoke... more Suspensions of self-propelled particles are studied in the framework of twodimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions.
We have recently developed two methods to calculate the magnetic normal modes of a magnetic nano-... more We have recently developed two methods to calculate the magnetic normal modes of a magnetic nano-particle. One of the methods is based on a conventional micromagnetic approach in which the time evolution of the magnetization of each cell is monitored. After filtering in frequency domain, the magnetic normal modes can be reconstructed. The second method is based on solving the same micromagneitc system in a dynamical matrix formulation. The results of the two methods, applied to a rectangular parallelepiped of Fe, will be presented and compared. r
In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (... more In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (HM) model which are then used as the basis for geometric discretizations of HM. Despite a topological obstruction to the existence of a global Lagrangian density, a local variational formulation allows one to derive local conservation laws using a version of Nöther's theorem from the formal variational calculus of Gelfand-Dikii. Using the local Lagrangian form we extend the method of Marsden, Patrick and Schkoller to derive local multisymplectic discretizations directly from the variational principle. We employ a version of the finite element method to discretize the space of sections of the trivial magnetic spin bundle N = M × S 2 over an appropriate space-time M . Since sections do not form a vector space, the usual FEM bases can be used only locally with coordinate transformations intervening on element boundaries, and conservation properties are guaranteed only within an element. We discuss possible ways of circumventing this problem, including the use of a local version of the method of characteristics, non-polynomial FEM bases and Lie-group discretization methods.
Eective viscosity (EV) of suspensions of puller-like microswimmers (pullers), such as the Chlamyd... more Eective viscosity (EV) of suspensions of puller-like microswimmers (pullers), such as the Chlamydamonas algae, is dicult to measure or to simulate for all swimmer concentrations. Although there are good reasons to expect that the EV of pullers is similar to that of passive suspensions, analytic determination of the passive EV for all concentrations remains unsatisfactory. At the same time, the EV of bacterial suspensions is closely linked to collective motion in these systems and is biologically signicant. We develop an approach to determining analytic EV estimates at all concentrations for suspensions of pullers as well as for passive suspensions.The proposed methods are based on the ideas of renormalization group theory (RG) and construct the EV formula based on the known asymptotics for small concentrations and near the critical point (i.e., approaching dense packing). For passive suspensions the method is veried by comparison against known theoretical results. We nd that the method performs much better than an earlier RG-based technique. For pullers the validation is done by comparing to experiments conducted on Chlamydamonas suspensions.
Recently an interesting new class of PDE integrators, multisymplectic schemes, has been introduce... more Recently an interesting new class of PDE integrators, multisymplectic schemes, has been introduced for solving systems possessing a certain multisymplectic structure. Some of the characteristic features of the method are its local nature (independent of boundary conditions) and an equal treatment of spatial and temporal variables. The nonlinear Schrödinger equation (NLS) has a multisymplectic formulation, and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the NLS. In the numerical experiments, the multisymplectic concatenated midpoint scheme (a centered cell discretization) is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff. On the other hand, an integrable Hamiltonian semi-discretization of NLS from Ablowitz and Ladik (AL) possesses a full set of global conservation laws and a noncanonical symplectic structure. We generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the AL type. Another approach examined in the paper is the introduction of transformations to reduce the AL system to either (1) separable form or (2) canonical form and then apply standard schemes in the new coordinates. All of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions. The performance of the schemes as well as interrelations among various geometric features are discussed.
We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbel... more We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by "pushing" or "pulling" both in three-and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria.
on three factors; the dipole moment of the coordinating ligands [1], the number of coordinating l... more on three factors; the dipole moment of the coordinating ligands [1], the number of coordinating ligands [2] and the cavity size of the coordination site [3]. By using free energy methods in molecular dynamics simulations to study model systems as well as a range of biological molecules including channels, transporters and enzymes, we are able to determine the contribution of each of these factors to the overall ion selectivity of the molecule. Varying contributions of each give rise to the richness in ion selectivity we see. By mapping out the importance of these factors in various cases, an estimation of the degree of selectivity of an ion selective biological molecule can be made given its structure. These results also assist in predicting the nature of a binding site of an unknown structure and have the potential to aid in the design of novel synthetic ion selective molecules.
A new representation of computational meshes is proposed in terms of a covering relation defined ... more A new representation of computational meshes is proposed in terms of a covering relation defined by discrete topological objects we call sieves. Fields over a mesh are handled locally by using the notion of refinement, dual to covering, and are later reassembled. In this approach fields are modeled by sections of a fiber bundle over a sieve. This approach cleanly separates the topology of the mesh from its geometry and other value-storage mechanisms. With these abstractions, finite element calculations are expressed using algorithms that are independent of mesh dimension, global topology, element shapes, and the finite element itself. Extensions and other applications are discussed. Despite what the name might suggest no extra relations between the elements of the chain are presumed. From the point of view of combinatorial topology we are dealing with chains mod 2 or chains with coefficients in the two-point field F 2 . However, we shall not use the algebraic structure or homology in this publication. 2 The obvious analogy with the categorical notions of product and coproduct will be clarified in our forthcoming paper .
Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous f... more Classical density functional theory ͑DFT͒ of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses O͑N log N͒ operations and O͑N͒ memory, where N is the number of grid points. This system-size scaling is significant because of the very large N required for three-dimensional systems. The algorithm uses fast Fourier transforms ͑FFTs͒ to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard ͑iterative substitution͒ iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT, we use fundamental measure theory. For the electrostatic DFT, we present two algorithms. One is for the "bulk-fluid" functional of Rosenfeld ͓Y. Rosenfeld, J. Chem. Phys. 98, 8126 ͑1993͔͒ that uses O͑N log N͒ operations. The other is for the "reference fluid density" ͑RFD͒ functional ͓D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 ͑2002͔͒. This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires O͑N 2 ͒ operations.
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Papers by Dmitry Karpeev