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32
FeaturePreserving Adaptive Mesh Generation for Molecular Shape Modeling and Simulation
, 2007
"... We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a n ..."
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Cited by 18 (7 self)
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We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: featurepreservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvment of meshes in a broad range of applications.
Topology, accuracy, and quality of isosurface meshes using dynamic particles
 IEEE Transactions on Visualization and Computer Graphics
, 2007
"... This paper describes a method for constructing isosurface triangulations of sampled, volumetric, threedimensional scalar fields. The resulting meshes consist of triangles that are of consistently high quality, making them well suited for accurate interpolation of scalar and vectorvalued quantities ..."
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Cited by 15 (3 self)
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This paper describes a method for constructing isosurface triangulations of sampled, volumetric, threedimensional scalar fields. The resulting meshes consist of triangles that are of consistently high quality, making them well suited for accurate interpolation of scalar and vectorvalued quantities, as required for numerous applications in visualization and numerical simulation. The proposed method does not rely on a local construction or adjustment of triangles as is done, for instance, in advancing wavefront or adaptive refinement methods. Instead, a system of dynamic particles optimally samples an implicit function such that the particles ’ relative positions can produce a topologically correct Delaunay triangulation. Thus, the proposed method relies on a global placement of triangle vertices. The main contributions of the paper are the integration of dynamic particles systems with surface sampling theory and PDEbased methods for controlling the local variability of particle densities, as well as detailing a practical method that accommodates Delaunay sampling requirements to generate sparse sets of points for the production of highquality tessellations. Index Terms—Isosurface extraction, particle systems, Delaunay triangulation.
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSONBOLTZMANN EQUATION
"... ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of t ..."
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Cited by 14 (8 self)
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ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the PoissonBoltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this twoterm regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L ∞ estimates to establish quasiorthogonality. To provide a highquality geometric model as input to the AFEM algorithm, we also describe a class of featurepreserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
Generating Triangulated Macromolecular Surfaces by Euclidean Distance Transform
, 2009
"... Macromolecular surfaces are fundamental representations of their threedimensional geometric shape. Accurate calculation of protein surfaces is of critical importance in the protein structural and functional studies including ligandprotein docking and virtual screening. In contrast to analytical or ..."
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Cited by 10 (0 self)
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Macromolecular surfaces are fundamental representations of their threedimensional geometric shape. Accurate calculation of protein surfaces is of critical importance in the protein structural and functional studies including ligandprotein docking and virtual screening. In contrast to analytical or parametric representation of macromolecular surfaces, triangulated mesh surfaces have been proved to be easy to describe, visualize and manipulate by computer programs. Here, we develop a new algorithm of EDTSurf for generating three major macromolecular surfaces of van der Waals surface, solventaccessible surface and molecular surface, using the technique of fast Euclidean Distance Transform (EDT). The triangulated surfaces are constructed directly from volumetric solids by a VertexConnected Marching Cube algorithm that forms triangles from grid points. Compared to the analytical result, the relative error of the surface calculations by EDTSurf is,2–4 % depending on the grid resolution, which is 1.5–4 times lower than the methods in the literature; and yet, the algorithm is faster and costs less computer memory than the comparative methods. The improvements in both accuracy and speed of the macromolecular surface determination should make EDTSurf a useful tool for the detailed study of protein docking and structure predictions. Both source code and the executable program of EDTSurf are freely available at
Adaptive Grid Based Methods for Computing Molecular Surfaces and Properties
, 2006
"... We present an adaptive grid based algorithm to compute a family of relevant molecular surfaces. Molecular interfaces are important in simulations and visualization involving biomolecules. The Richards surface has traditionally been used as a good approximation to the surface, and defined as the surf ..."
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Cited by 9 (3 self)
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We present an adaptive grid based algorithm to compute a family of relevant molecular surfaces. Molecular interfaces are important in simulations and visualization involving biomolecules. The Richards surface has traditionally been used as a good approximation to the surface, and defined as the surface formed by the inner facing part of a solvent probe atom rolling along the van der Waals surface of the molecule. Computing and representing this surface has traditionally involved complex geometrical data structures like alpha shapes. Adaptive and uniform trilinear grids are commonly used in various simulations involving interactions of molecules or computation of electrostatics and other energy terms. We make use of this grid directly to compute the Molecular Surface and properties like area, volume, curvatures, surface atoms and other surfaces. We compare geometrical and biochemical properties with other methods as a validation. 1 Molecular Surface Definitions Explicit surface definitions as the interface between the solvent and proteins have been given since 1970s. Since it is easier to handle implicitly defined models mathematically, different implicit approximations to these surfaces have been developed. 1.1 van der Waals and Lee Richards Surface Definitions The most common model for molecules is as a collection of atoms represented by spheres, with radii equal to their van der Waals radii. The surface of the set of spheres is known as the van der Waals surface. Lee and Richards introduced the concept
An automatic 3D mesh generation method for domains with multiple materials. Computer methods in applied mechanics and engineering
, 2010
"... This paper describes an automatic and efficient approach to construct unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials. The boundaries of these material regions form nonmanifold surfaces. In earlier papers, we developed an octreebased isocon ..."
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Cited by 8 (0 self)
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This paper describes an automatic and efficient approach to construct unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials. The boundaries of these material regions form nonmanifold surfaces. In earlier papers, we developed an octreebased isocontouring method to construct unstructured 3D meshes for a singlematerial (homogeneous) domain with manifold boundary. In this paper, we introduce the notion of a material change edge and use it to identify the interface between two or several different materials. A novel method to calculate the minimizer point for a cell shared by more than two materials is provided, which forms a nonmanifold node on the boundary. We then mesh all the material regions simultaneously and automatically while conforming to their boundaries directly from volumetric data. Both material change edges and interior edges are analyzed to construct tetrahedral meshes, and interior grid points are analyzed for proper hexahedral mesh construction. Finally, edgecontraction and smoothing methods are used to improve the quality of tetrahedral meshes, and a combination of pillowing, geometric flow and optimization techniques is used for hexahedral mesh quality improvement. The shrink set of pillowing schemes is defined automatically as the boundary of each material region. Several application results of our multimaterial mesh generation method are also provided. Key words: Unstructured 3D meshes, multiple materials, conforming boundaries, material change edge, pillowing, geometric flow. 1
Smooth surface constructions via a higher order level set method
, 2007
"... We present a general framework for a higherorder spline levelset (HLS) method and apply this to smooth surface constructions. Starting from a first order energy functional, we derive the general level set formulation, and provide an efficient solution of a second order geometric partial differenti ..."
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Cited by 8 (5 self)
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We present a general framework for a higherorder spline levelset (HLS) method and apply this to smooth surface constructions. Starting from a first order energy functional, we derive the general level set formulation, and provide an efficient solution of a second order geometric partial differential equation using a C² spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Ztransform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. We provide two demonstrative smooth surface construction examples of our HLS method. The first is the construction of a smooth surface model (an implicit solvation interface) of biomolecules in solvent, given their individual atomic coordinates and solvated radii. The second is the smooth surface reconstruction from a cloud of points generated from a 3D surface scanner.
HighFidelity Geometric Modeling for Biomedical Applications
"... We describe a combination of algorithms for high fidelity geometric modeling and mesh generation. Although our methods and implementations are applicationneutral, our primary target application is multiscale biomedical models that range in scales across the molecular, cellular, and organ levels. Ou ..."
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Cited by 8 (5 self)
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We describe a combination of algorithms for high fidelity geometric modeling and mesh generation. Although our methods and implementations are applicationneutral, our primary target application is multiscale biomedical models that range in scales across the molecular, cellular, and organ levels. Our software toolchain implementing these algorithms is general in the sense that it can take as input a molecule in PDB/PQR forms, a 3D scalar volume, or a userdefined triangular surface mesh that may have very low quality. The main goal of our work presented is to generate high quality and smooth surface triangulations from the aforementioned inputs, and to reduce the mesh sizes by mesh coarsening. Tetrahedral meshes are also generated for finite element analysis in biomedical applications. Experiments on a number of biostructures are demonstrated, showing that our approach possesses several desirable properties: featurepreservation, local adaptivity, high quality, and smoothness (for surface meshes). The availability of this software toolchain will give researchers in computational biomedicine and other modeling areas access to higherfidelity geometric models.
FAST MOLECULAR SOLVATION ENERGETICS AND FORCE COMPUTATION ∗
"... Abstract. The total free energy of a molecule includes the classical molecular mechanical energy (which is understood as the free energy in vacuum) and the solvation energy which is caused by the change of the environment of the molecule (solute) from vacuum to solvent. The solvation energy is impor ..."
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Cited by 5 (2 self)
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Abstract. The total free energy of a molecule includes the classical molecular mechanical energy (which is understood as the free energy in vacuum) and the solvation energy which is caused by the change of the environment of the molecule (solute) from vacuum to solvent. The solvation energy is important to the study of the intermolecular interactions. In this paper we develop a fast surfacebased generalized Born method to compute the electrostatic solvation energy along with the energy derivatives for the solvation forces. The most timeconsuming computation is the evaluation of the surface integrals over an algebraic spline molecular surface (ASMS) and the fast computation is achieved by the use of the nonequispaced fast Fourier transform (NFFT) algorithm. The main results of this paper involve (a) an efficient sampling of quadrature points over the molecular surface by using nonlinear patches, (b) fast linear time estimation of energy and intermolecular forces, (c) error analysis, and (d) efficient implementation combining fast pairwise summation and the continuum integration using nonlinear patches.
A fast variational method for the construction of resolution adaptive C²smooth molecular surfaces
 COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
, 2009
"... We present a variational approach to smooth molecular (proteins, nucleic acids) surface constructions, starting from atomic coordinates, as available from the protein and nucleicacid data banks. Molecular dynamics (MD) simulations traditionally used in understanding protein and nucleicacid folding ..."
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Cited by 4 (1 self)
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We present a variational approach to smooth molecular (proteins, nucleic acids) surface constructions, starting from atomic coordinates, as available from the protein and nucleicacid data banks. Molecular dynamics (MD) simulations traditionally used in understanding protein and nucleicacid folding processes, are based on molecular force fields, and require smooth models of these molecular surfaces. To accelerate MD simulations, a popular methodology is to employ coarse grained molecular models, which represent clusters of atoms with similar physical properties by psuedo atoms, resulting in coarser resolution molecular surfaces. We consider generation of these mixedresolution or adaptive molecular surfaces. Our approach starts from deriving a general form second order geometric partial differential equation in the levelset formulation, by minimizing a first order energy functional which additionally includes a regularization term to minimize the occurrence of chemically infeasible molecular surface pockets or tunnellike artifacts. To achieve even higher computational efficiency, a fast cubic Bspline C² interpolation algorithm is also utilized. A narrow band, tricubic Bspline levelset method is then used to provide C² smooth and resolution adaptive molecular surfaces.