arXiv subjects

N. Sukumar

Publications and source records attributed to N. Sukumar.

33 records · Page 2Linked to original sources

arXiv2022

Virtual element method for modeling the deformation of multiphase composites

In this paper, we study applications of the virtual element method (VEM) for simulating the deformation of multiphase composites. The VEM is a Galerkin approach that is applicable to meshes that consist of arbitrarily-shaped polygonal and polyhedral (simple and nonsimple) elements. In the VEM, the basis functions are defined as the solution of a local elliptic partial differential equation, and are never explicitly computed in the implementation of the method. The stifness matrix of each element is built by using the elliptic projection operator of the internal virtual work (bilinear form) and it consists of two terms: a consistency term that is exactly computed (linear patch test is satisfied) and a correction term (ensures stability) that is orthogonal to affine displacement fields and has the right scaling. The VEM simplifies mesh generation for a multiphase composite: a stiff inclusion can be modeled using a single polygonal or polyhedral element. Attributes of the virtual element approach are highlighted through comparisons with Voronoi-cell lattice models, which provide discrete representations of material structure. The comparisons involve a suite of two-dimensional linear elastic problems: patch test, axisymmetric circular inclusion problem, and the deformation of a three-phase composite. The simulations demonstrate the accuracy and flexibility of the virtual element method.

math.NA↗

arXiv2021

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artifical neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct $ϕ$, an approximate distance function to the boundary of a domain. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as $ϕ$ multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. We present numerical solutions for linear and nonlinear boundary-value problems over domains with affine and curved boundaries. Benchmark problems in 1D for linear elasticity, advection-diffusion, and beam bending; and in 2D for the Poisson equation, biharmonic equation, and the nonlinear Eikonal equation are considered. The approach extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the 4D hypercube. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

math.NA↗

arXiv2021

Extended virtual element method for two-dimensional linear elastic fracture

In this paper, we propose an eXtended Virtual Element Method (X-VEM) for two-dimensional linear elastic fracture. This approach, which is an extension of the standard Virtual Element Method (VEM), facilitates mesh-independent modeling of crack discontinuities and elastic crack-tip singularities on general polygonal meshes. For elastic fracture in the X-VEM, the standard virtual element space is augmented by additional basis functions that are constructed by multiplying standard virtual basis functions by suitable enrichment fields, such as asymptotic mixed-mode crack-tip solutions. The design of the X-VEM requires an extended projector that maps functions lying in the extended virtual element space onto a set spanned by linear polynomials and the enrichment fields. An efficient scheme to compute the mixed-mode stress intensity factors using the domain form of the interaction integral is described. The formulation permits integration of weakly singular functions to be performed over the boundary edges of the element. Numerical experiments are conducted on benchmark mixed-mode linear elastic fracture problems that demonstrate the sound accuracy and optimal convergence in energy of the proposed formulation.

math.NA↗

arXiv2021

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces integration over a region bounded by $m$ curves to integration over $m$ regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. This paper also introduces several methods for smoothing integrands with point singularities and near-singularities. When these methods are used, highly efficient integration of weakly singular functions is realized. The SBC method is applied to a number of benchmark problems, which reveal its broad applicability and superior performance (in terms of time to generate a rule and accuracy per cubature point) when compared to existing methods for integration.

math.NA↗

arXiv2020

Spectral extended finite element method for band structure calculations in phononic crystals

In this paper, we compute the band structure of one- and two-dimensional phononic composites using the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes. On using partition-of-unity enrichment in finite element analysis, the X-FEM permits use of structured finite element meshes that do not conform to the geometry of holes and inclusions. This eliminates the need for remeshing in phononic shape optimization and topology optimization studies. In two dimensions, we adopt rational B{é}zier representation of curved (circular) geometries, and construct suitable material enrichment functions to model two-phase composites. A Bloch-formulation of the elastodynamic phononic eigenproblem is adopted. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme -- a method that uses Euler's homogeneous function theorem and Stokes's theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used on finite elements that are cut by a hole. Band structure calculations on perforated (circular holes, elliptical holes, and holes defined as a level set) materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes. Several numerical examples demonstrate the advantages of $p$-refinement made possible by the spectral extended finite element method. In these examples, fourth-order spectral extended finite elements deliver $\mathcal{O}(10^{-8})$ accuracy in frequency calculations with more than thirty-fold fewer degrees-of-freedom when compared to quadratic finite elements.

physics.comp-ph↗

arXiv2019

A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition

In this paper, we present a novel nodal integration scheme for meshfree Galerkin methods that draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the bilinear form is decomposed into a consistency part and a stability part that ensure consistency and stability of the method. The performance of the proposed nodal integration scheme is assessed through benchmark problems in linear and nonlinear analyses of solids for small displacements and small-strain kinematics. Numerical results are presented for linear elastostatics and linear elastodynamics, and viscoelasticity. We demonstrate that the proposed nodally integrated meshfree method is accurate, converges optimally, and is more reliable and robust than a standard cell-based Gauss integrated meshfree method.

math.NA↗

arXiv2019

Smallest Fullerene-like Structures of Boron with Cr, Mo, and W Encapsulation

Using density functional theory calculations, we study doping of a Cr, Mo, and W atom in boron clusters in the size range of 18-24 atoms and report the finding of metal atom encapsulated fullerene-like cage structures with 20 to 24 boron atoms in contrast to a fullerene-like structure of pure boron with 40 atoms. Our results show that bicapped drum-shaped structures are favored for neutral Cr@B$_{18}$, Mo@B$_{20}$, and W@B$_{20}$ clusters whereas a drum-shaped structure is preferred for neutral, cation, and anion of Mo@B$_{18}$ and W@B$_{18}$. Further, we find that B$_{20}$ is the smallest cage for Cr encapsulation, while B$_{22}$ is the smallest symmetric cage for Mo and W encapsulation and it is magic. Symmetric cage structures are also obtained for Mo@B$_{24}$ and W@B$_{24}$. A detailed analysis of the bonding character and molecular orbitals suggests that Cr@B$_{18}$, Cr@B$_{20}$, M@B$_{22}$ (M = Cr, Mo, and W) and M@B$_{24}$ (M = Mo and W) cages are stabilized with 18 $π$-bonded valence electrons whereas the drum-shaped M@B$_{18}$ (M = Mo and W) clusters are stabilized by 20 $π$-bonded valence electrons. Calculations with PBE0 functional in Gaussian 09 code show that in all cases of neutral clusters there is a large highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap. In some cases the lowest energy isomer of the charged clusters is different from the one for the neutral. We discuss the calculated infrared and Raman spectra for the neutral and cation clusters as well as the electronic structure of the anion clusters. Also we report results for isoelectronic anion and neutral clusters doped with V, Nb, and Ta which are generally similar to those obtained for Mo and W doped clusters. These results would be helpful to confirm the formation of these doped boron clusters experimentally.

physics.atm-clus↗

arXiv2018

Orbital-enriched Flat-top Partition of Unity Method for the Schrödinger Eigenproblem

Quantum mechanical calculations require the repeated solution of a Schrödinger equation for the wavefunctions of the system. Recent work has shown that enriched finite element methods significantly reduce the degrees of freedom required to obtain accurate solutions. However, time to solution has been adversely affected by the need to solve a generalized eigenvalue problem and the ill-conditioning of associated systems matrices. In this work, we address both issues by proposing a stable and efficient orbital-enriched partition-of-unity method to solve the Schrödinger boundary-value problem in a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In our proposed PUM, the three-dimensional domain is covered by overlapping patches, with a compactly-supported, non-negative weight function, that is identically equal to unity over some finite subset of its support associated with each patch. This so-called flat-top property provides a pathway to devise a stable approximation over the whole domain. On each patch, we use $p$-th degree orthogonal polynomials that ensure $p$-th order completeness, and in addition include eigenfunctions of the radial solution of the Schrödinger equation. Furthermore, we adopt a variational lumping approach to construct a block-diagonal overlap matrix that yields a standard eigenvalue problem and demonstrate accuracy, stability and efficiency of the method.

physics.comp-ph↗

arXiv2016

Partition of unity finite element method for quantum mechanical materials calculations

The current state of the art for large-scale quantum-mechanical simulations is the planewave (PW) pseudopotential method, as implemented in codes such as VASP, ABINIT, and many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points in space, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires significant nonlocal communications, which limit parallel efficiency. Real-space methods such as finite-differences and finite-elements have partially addressed both resolution and parallel-communications issues but have been plagued by one key disadvantage relative to PW: excessive number of degrees of freedom needed to achieve the required accuracies. We present a real-space partition of unity finite element (PUFE) method to solve the Kohn-Sham equations of density functional theory. In the PUFE method, we build the known atomic physics into the solution process using partition-of-unity enrichment techniques in finite element analysis. The method developed herein is completely general, applicable to metals and insulators alike, and particularly efficient for deep, localized potentials, as occur in calculations at extreme conditions of pressure and temperature. Full self-consistent Kohn-Sham calculations are presented for LiH, involving light atoms, and CeAl, involving heavy atoms with large numbers of atomic-orbital enrichments. We find that the new PUFE approach attains the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the PW method. We compute the equation of state of LiH and show that the computed lattice constant and bulk modulus are in excellent agreement with reference PW results, while requiring an order of magnitude fewer degrees of freedom to obtain.

physics.comp-ph↗

arXiv2016

Convergence analysis of a locally accelerated preconditioned steepest descent method for Hermitian-definite generalized eigenvalue problems

By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of the rate of convergence of the \psdid method. We show that with the proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the \psdid method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper sheds light on an improved understanding of the convergence behavior of these block methods.

math.NA↗

arXiv2013

Gradient bounds for Wachspress coordinates on polytopes

We derive upper and lower bounds on the gradients of Wachspress coordinates defined over any simple convex d-dimensional polytope P. The bounds are in terms of a single geometric quantity h_*, which denotes the minimum distance between a vertex of P and any hyperplane containing a non-incident face. We prove that the upper bound is sharp for d=2 and analyze the bounds in the special cases of hypercubes and simplices. Additionally, we provide an implementation of the Wachspress coordinates on convex polyhedra using Matlab and employ them in a 3D finite element solution of the Poisson equation on a non-trivial polyhedral mesh. As expected from the upper bound derivation, the H^1-norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.

math.NA↗

arXiv2013

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods.

cond-mat.mtrl-sci↗

arXiv2012

Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.

math.NA↗

arXiv2011

Linear scaling solution of the all-electron Coulomb problem in solids

We present a linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. The resulting method is systematically improvable and well suited to large-scale quantum mechanical calculations in which the Coulomb potential and energy of a continuous electronic density and singular nuclear density are required. Linear scaling is achieved by introducing smooth, strictly local neutralizing densities to render nuclear interactions strictly local, and solving the remaining neutral Poisson problem for the electrons in real space. While the formulation includes singular nuclear potentials without smearing approximations, the required Poisson solution is in Sobolev space $H^1$, as required for convergence in the energy norm. We employ enriched finite elements, with enrichments from isolated atom solutions, for an efficient solution of the resulting Poisson problem in the interacting solid. We demonstrate the accuracy and convergence of the approach by direct comparison to standard Ewald sums for a lattice of point charges, and demonstrate the accuracy in all-electron quantum mechanical calculations with an application to crystalline diamond.

cond-mat.mtrl-sci↗

arXiv1998

Effect of Boundary Conditions on Cellular Automata that Classify Density

The properties of two-state nearest-neighbour cellular automata (CA) that are capable of density classification are discussed. It is shown that these CA actually conserve the total density, rather than merely classifying it. This is also the criterion for any CA simulation of DFT. The effect of boundary and periodicity conditions upon the evolution of such CA are elaborated by considering linear and cyclic lattices and boundary conditions consistent with the conservation criterion. In a bounded linear lattice, it is possible to achieve a configuration with a single 01 and no 10 domain wall, or vice versa, but this is not possible in a cyclic lattice, where these two domain walls have to appear in pairs. This determines the final stable state of the automaton.

comp-gas↗