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Nathan A Baker - One of the best experts on this subject based on the ideXlab platform.
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Finite Element Solution of the steady state smoluchowski equation for rate constant calculations
Biophysical Journal, 2004Co-Authors: Yuhua Song, Yongjie Zhang, Tongye Shen, Chandrajit L Bajaj, Andrew J Mccammon, Nathan A BakerAbstract:This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, Finite Element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these Finite Element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.
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the adaptive multilevel Finite Element Solution of the poisson boltzmann equation on massively parallel computers
Ibm Journal of Research and Development, 2001Co-Authors: Nathan A Baker, Michael Holst, David Sept, J A MccammonAbstract:By using new methods for the parallel Solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel Finite Element Solution of the Poisson-Boltzmann equation for a microtubule on the NPACI Blue Horizon--a massively parallel IBM RS/6000® SP with eight POWER3 SMP nodes. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600000 atoms, and has a net charge of -1800 e. Poisson-Boltzmann calculations are performed for several processor configurations, and the algorithm used shows excellent parallel scaling.
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adaptive multilevel Finite Element Solution of the poisson boltzmann equation i algorithms and examples
Journal of Computational Chemistry, 2000Co-Authors: Michael Holst, Nathan A Baker, Feng WangAbstract:This article is the first of two articles on the adaptive multilevel Finite Element treatment of the nonlinear Poisson–Boltzmann equation (PBE), a nonlinear eliptic equation arising in biomolecular modeling. Fast and accurate numerical Solution of the PBE is usually difficult to accomplish, due to the presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domain, and rapid (exponential) nonlinearity. In this first article, we explain how adaptive multilevel Finite Element methods can be used to obtain extremely accurate Solutions to the PBE with very modest computational resources, and we present some illustrative examples using two well-known test problems. The PBE is first discretized with piece-wise linear Finite Elements over a very coarse simplex triangulation of the domain. The resulting nonlinear algebraic equations are solved with global inexact Newton methods, which we have described in an article appearing previously in this journal. A posteriori error estimates are then computed from this discrete Solution, which then drives a simplex subdivision algorithm for performing adaptive mesh refinement. The discretize–solve–estimate–refine procedure is then repeated, until a nearly uniform Solution quality is obtained. The sequence of unstructured meshes is used to apply multilevel methods in conjunction with global inexact Newton methods, so that the cost of solving the nonlinear algebraic equations at each step approaches optimal O(N) linear complexity. All of the numerical procedures are implemented in MANIFOLD CODE (MC), a computer program designed and built by the first author over several years at Caltech and UC San Diego. MC is designed to solve a very general class of nonlinear elliptic equations on complicated domains in two and three dimensions. We describe some of the key features of MC, and give a detailed analysis of its performance for two model PBE problems, with comparisons to the alternative methods. It is shown that the best available uniform mesh-based Finite difference or box-method algorithms, including multilevel methods, require substantially more time to reach a target PBE Solution accuracy than the adaptive multilevel methods in MC. In the second article, we develop an error estimator based on geometric solvent accessibility, and present a series of detailed numerical experiments for several complex biomolecules. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1319–1342, 2000
Weizhang Huang - One of the best experts on this subject based on the ideXlab platform.
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an adaptive moving mesh Finite Element Solution of the regularized long wave equation
Journal of Scientific Computing, 2018Co-Authors: Weizhang Huang, Jianxian QiuAbstract:An adaptive moving mesh Finite Element method is proposed for the numerical Solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical Solution of those equations, a $$C^0$$ Finite Element method cannot apply directly on a moving mesh since the mixed derivatives of the Finite Element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear Finite Elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train Solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the Solution.
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a study on moving mesh Finite Element Solution of the porous medium equation
Journal of Computational Physics, 2017Co-Authors: Weizhang HuangAbstract:An adaptive moving mesh Finite Element method is studied for the numerical Solution of the porous medium equation with and without variable exponents and absorption. The method is based on the moving mesh partial differential equation approach and employs its newly developed implementation. The implementation has several improvements over the traditional one, including its explicit, compact form of the mesh velocities, ease to program, and less likelihood of producing singular meshes. Three types of metric tensor that correspond to uniform and arclength-based and Hessian-based adaptive meshes are considered. The method shows first-order convergence for uniform and arclength-based adaptive meshes, and second-order convergence for Hessian-based adaptive meshes. It is also shown that the method can be used for situations with complex free boundaries, emerging and splitting of free boundaries, and the porous medium equation with variable exponents and absorption. Two-dimensional numerical results are presented.
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an anisotropic mesh adaptation method for the Finite Element Solution of heterogeneous anisotropic diffusion problems
Journal of Computational Physics, 2010Co-Authors: Xianping Li, Weizhang HuangAbstract:Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical Solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous Solution. A well known mesh condition for the DMP satisfaction by the linear Finite Element Solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh Elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the Finite Element Solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear Finite Element method for anisotropic meshes generated with the metric tensors.
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an anisotropic mesh adaptation method for the Finite Element Solution of variational problems
Finite Elements in Analysis and Design, 2010Co-Authors: Weizhang HuangAbstract:It has been amply demonstrated that anisotropic mesh adaptation can significantly improve computational efficiency over isotropic mesh adaptation especially for problems with strong anisotropic features. Although numerous research has been done on isotropic mesh adaptation for Finite Element Solution of variational problems, little work has been done on anisotropic mesh adaptation. In this paper we consider anisotropic mesh adaptation method for the Finite Element Solution of variational problems. A bound for the first variation of a general functional is derived, which is semi-a posteriori in the sense that it involves the residual and edge jump, both dependent on the computed Solution, as well as the Hessian of the exact Solution. A formula for the metric tensor M for use in anisotropic mesh adaptation is defined such that the bound is minimized on a mesh that is uniform in the metric specified by M (i.e., an M-uniform mesh). Interestingly, when restricted to isotropic meshes, we can obtain a similar but completely a posteriori bound and the corresponding formula for the metric tensor. When M is defined, an anisotropic adaptive mesh is generated as an M-uniform mesh. Numerical results demonstrate that the new mesh adaptation method is comparable in performance with existing ones based on interpolation error and has the advantage that the resulting mesh also adapts to changes in the structure of the underlying problem.
Panayiotis Papadopoulos - One of the best experts on this subject based on the ideXlab platform.
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on the Finite Element Solution of frictionless contact problems using an exact penalty approach
Computer Methods in Applied Mechanics and Engineering, 2020Co-Authors: Fabian Sewerin, Panayiotis PapadopoulosAbstract:Abstract In this article, an exact penalty method is introduced for the enforcement of the impenetrability constraint in problems of frictionless two-body contact solved by the Finite Element method. A complete algorithmic implementation is presented, including an automated scheme for the selection of the penalty parameter. Numerical examples are employed to assess the accuracy and robustness of the exact penalty method in comparison to classical penalty and Lagrange multiplier alternatives.
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a lagrange multiplier method for the Finite Element Solution of frictionless contact problems
Mathematical and Computer Modelling, 1998Co-Authors: Panayiotis Papadopoulos, Jerome M SolbergAbstract:This article proposes a novel Lagrange multiplier-based formulation for the Finite Element Solution of the quasi-static two-body contact problem in the presence of Finite motions and deformations. The main idea rests in the interpretation of the two-body contact as a composition of two simultaneous Signorini-like problems, which naturally yield geometrically unbiased approximations of the kinematics and kinetics of frictionless contact. A two-dimensional Finite Element is introduced that exactly satisfies the impenetrability constraint and allows for the direct computation of consistent pressure distributions on the interacting surfaces.
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a mixed formulation for the Finite Element Solution of contact problems
Applied Mechanics and Engineering, 1992Co-Authors: Panayiotis Papadopoulos, Robert L TaylorAbstract:Abstract In this paper we present a Finite Element algorithm for the static Solution of two-dimensional frictionless contact problems involving bodies undergoing arbitrarily large motions and deformations. A mixed penalty formulation is employed in approximating the resulting variational inequalities. The algorithm is applied to quadratic Elements along with a rational scheme for determining the contacting regions. Several numerical simulations illustrate the applicability and accuracy of the proposed Solution procedure.
Hongbin Zhan - One of the best experts on this subject based on the ideXlab platform.
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a Finite Element Solution for the fractional advection dispersion equation
Advances in Water Resources, 2008Co-Authors: Quanzhong Huang, Guanhua Huang, Hongbin ZhanAbstract:Abstract The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable Finite Element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM Solution is verified against the analytical Solution, and the sensitivity of the FEM Solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical Solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM Solution of this study has smaller numerical dispersion than that of the FD Solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α , the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM Solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves.
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a Finite Element Solution for the fractional advection dispersion equation
Advances in Water Resources, 2008Co-Authors: Quanzhong Huang, Guanhua Huang, Hongbin ZhanAbstract:Abstract The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable Finite Element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM Solution is verified against the analytical Solution, and the sensitivity of the FEM Solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical Solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM Solution of this study has smaller numerical dispersion than that of the FD Solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α, the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM Solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves.
Peter Hansbo - One of the best experts on this subject based on the ideXlab platform.
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least squares stabilized augmented lagrangian multiplier method for elastic contact
Finite Elements in Analysis and Design, 2016Co-Authors: Peter Hansbo, Asim Rashid, Kent SalomonssonAbstract:In this paper, we propose a stabilized augmented Lagrange multiplier method for the Finite Element Solution of small deformation elastic contact problems. We limit ourselves to friction-free contact with a rigid obstacle, but the formulation is readily extendable to more complex situations. HighlightsStabilized method for Finite Element Solution of elastic contact problems.Stabilized augmented Lagrange multiplier method for contact.FEA Solution of Hertzian contact problem.
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Interior-penalty-stabilized Lagrange multiplier methods for the Finite-Element Solution of elliptic interface problems
IMA Journal of Numerical Analysis, 2009Co-Authors: Erik Burman, Peter HansboAbstract:In this paper we propose a class of jump-stabilized Lagrange multiplier methods for the Finite-Element Solution of multidomain elliptic partial differential equations using piecewise-constant or continuous piecewise-linear approximations of the multipliers. For the purpose of stabilization we use the jumps in derivatives of the multipliers or, for piecewise constants, the jump in the multipliers themselves, across Element borders. The ideas are illustrated using Poisson's equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.
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a lagrange multiplier method for the Finite Element Solution of elliptic interface problems using non matching meshes
Numerische Mathematik, 2005Co-Authors: Peter Hansbo, C Lovadina, Ilaria Perugia, Giancarlo SangalliAbstract:In this paper we propose a Lagrange multiplier method for the Finite Element Solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poisson’s equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.
