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The Numerical Approximation of a Delta Function with Application to Level Set Methods
Abstract
It is shown that a discrete delta function can be constructed using a technique developed by Anita Mayo [The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Sci. Comput. 21 (1984) 285–299] for the numerical solution of elliptic equations with discontinuous source terms. This delta function is concentrated on the zero level set of a continuous function. In two space dimensions, this corresponds to a line and a surface in three space dimensions. Delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function. The numerical implementation of these delta functions achieves the expected order of accuracy.
... From the level set grid function φ i , one typically approximates the curve or surface Γ 0 by Γ , (generally a polygon), and then uses this approximation to compute the integral of f over Γ 0 using local parameterizations of Γ (see e.g. [9,18,21]). Recently in [17]), it is proposed to convert the implicit geometry into the graph of an implicitly given height function, leading to a recursive algorithm on the number of dimensions and thus requiring only one-dimensional rootfinding and one-dimensional Gaussian quadratures. ...
... 2. Derive an analytical integral formulation I(f, φ) that is easy to discretize, then discretize it (see e.g. [21,22,28,25,27,26,7,8]). We note that this approach computes the integral (2) without using any local parameterizations of Γ 0 . ...
... One choice is to take independent of the level set function φ and the grid. In the work of Smereka [21], the discrete delta-function is concentrated within one grid cell on either side of the interface, and is obtained by discretizing the fundamental solution of the Laplace equation using ghost-points. In the work of Towers [22], the discretized delta function is computed via two different formulations involving the Heaviside function. ...
We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g. curves in two dimensions that contain finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.
... This is equivalent to the elliptic interface problem: Δu = 0 with [u] = 0 and [∇u · n] = 1. This connection is used to construct a second-order accurate discrete approximation of the delta function in [63]. ...
... We propose what can be considered a hybrid that we call the Compact Coupling Interface Method (CCIM). Our method combines elements of CIM [22] and Smereka's work on second-order accurate discrete delta functions by setting up an elliptic interface problem with interfacial jump conditions [63]. The use of Smereka's setup, itself based on Mayo's work in [64], allows us to remove the quadratic polynomial approximations of CIM2 and its need for two points on either side of the interface in a direction that crosses the interface, thus compacting the stencil and allowing more applicability in generating accurate princi- Fig. 2 Examples of a (1) CIM2 stencil and a (2) CCIM stencil at x i . ...
... This section follows the derivation found in Smereka's work [63]. Along the coordinate direction e k , if the interface does not intersect the grid segment x i x i+e k , then by Taylor's theorem, ...
We propose the Compact Coupling Interface Method, a finite difference method capable of obtaining second-order accurate approximations of not only solution values but their gradients, for elliptic complex interface problems with interfacial jump conditions. Such elliptic interface boundary value problems with interfacial jump conditions are a critical part of numerous applications in fields such as heat conduction, fluid flow, materials science, and protein docking, to name a few. A typical example involves the construction of biomolecular shapes, where such elliptic interface problems are in the form of linearized Poisson–Boltzmann equations, involving discontinuous dielectric constants across the interface, that govern electrostatic contributions. Additionally, when interface dynamics are involved, the normal velocity of the interface might be comprised of the normal derivatives of solution, which can be approximated to second-order by our method, resulting in accurate interface dynamics. Our method, which can be formulated in arbitrary spatial dimensions, combines elements of the highly-regarded Coupling Interface Method, for such elliptic interface problems, and Smereka’s second-order accurate discrete delta function. The result is a variation and hybrid with a more compact stencil than that found in the Coupling Interface Method, and with advantages, borne out in numerical experiments involving both geometric model problems and complex biomolecular surfaces, in more robust error profiles.
... Enquist et al. [10] later presented discretization techniques for the Dirac delta that were first order accurate with a second order convergence rate. In [11], Smereka further developed a second order accurate discretization of the Dirac delta function with application to calculating line and surface integrals in level set based methods. ...
... They showed the approach to be second-order accurate and being robust to interface perturbation on the grid. They showed several practical examples and compared their results to the techniques introduced by Smereka in [11]. ...
... Here we compare our method to the Geometric numerical integration method of Min-Gibou [15]. In [15], the authors had compared their Geometric numerical integration method to that of the first-order and second-order Delta function formulation [11]. Comparisons were made on accounts of robustness to perturbations of the interface location on the grid, order of convergence and numerical accuracy. ...
If we wish to integrate a function along a single T-level surface of a function , then a number of different methods for extracting finite elements appropriate to the dimension of the level surface may be employed to obtain an explicit representation over which the integration may be performed using standard numerical quadrature techniques along each element. However, when the goal is to compute an entire continuous family m(T) of integrals over all the T-level surfaces of , then this method of explicit level set extraction is no longer practical. We introduce a novel method to perform this type of numerical integration efficiently by making use of the coarea formula. We present the technique for discretization of the coarea formula and present the algorithms to compute the integrals over families of T-level surfaces. While validation of our method in the special case of a single level surface demonstrates accuracies close to more explicit isosurface integration methods, we show a sizable boost in computational efficiency in the case of multiple T-level surfaces, where our coupled integration algorithms significantly outperform sequential one-at-a-time application of explicit methods.
... (2.11)) using a standard sampling technique such as the midpoint rule [10]. [28], Smereka (2006) [29]. Other approaches have been investigated to compute the interfacial area with the signed distance function. ...
... (2.11)) using a standard sampling technique such as the midpoint rule [10]. [28], Smereka (2006) [29]. Other approaches have been investigated to compute the interfacial area with the signed distance function. ...
... with 500 Θ i = min (θ (ψ ij )) , j ∈ neigh(i), (4. 29) and ...
A new method is developed to approximate a first-order Hamilton–Jacobi equation. The constant motion of an interface in the normal direction is of interest. The interface is captured with the help of a “Level-Set” function approximated through a finite-volume Godunov-type scheme. Contrarily to most computational approaches that consider smooth Level-Set functions, the present one considers sharp “Level-Set”, the numerical diffusion being controlled with the help of the Overbee limiter (Chiapolino et al. in J Comput Phys 340:389–417, 2017). The method requires gradient computation that is addressed through the least squares approximation. Multidimensional results on fixed unstructured meshes are provided and checked against analytical solutions. Geometrical properties such as interfacial area and volume computation are addressed as well. Results show excellent agreement with the exact solutions.
... Surface integrals can be evaluated similarly to (42): One must only replace all computations concerning volumes with surface equivalents and further Ξ (S) with T (Υ c (S)) because these cut points span the reconstructed plane V θ ∩ {Φ h = 0}. For more details about surface reconstruction in the context of finite elements, we refer to [41,42], and for integration approaches beyond this one [13,43]. ...
... For the simulations we consider bi-or trilinear shape functions. In all cases we applied a preconditioned conjugate gradient (CG) method in order to solve (43). As preconditioner we used the factor from an incomplete Cholesky decomposition of the matrix on the left-hand side of (43). ...
... In all cases we applied a preconditioned conjugate gradient (CG) method in order to solve (43). As preconditioner we used the factor from an incomplete Cholesky decomposition of the matrix on the left-hand side of (43). As drop tolerance we chose 10 −4 and 10 −3 for two and three dimensional problems, respectively. ...
Numerical methods operating on structured grids have become popular since they offer good run time performance and are able to process directly voxel-based digital data from image recordings. Hence, the general framework of these fast solvers presupposes an unfitted boundary approximation avoiding complicated meshing of bodies. This allows an efficient handling with geometrical issues. Nevertheless, contacts between deformable solids are hard to deal with in the presence of this boundary representation. For this difficulty we suggest the usage of an implicit boundary representation combined with a modified saddle point formulation, resembling Nitsche's approach. Both ideas give an elegant approach for discretizing the contact terms and enable a simple contact detection. Moreover, we suggest an intermediate surface as new reference contact area which fits well into our proposed method. In the end we present numerical results and analyze the accuracy and convergence rate. Furthermore, we demonstrate the current application range of our approach for problems with multiple contacts. In this paper we focus only on frictionless contact with small deformations.
... From the level set grid function φ i , one typically approximates the curve or surface Γ 0 by Γ , (generally a polygon), and then uses this approximation to compute the integral of f over Γ 0 using local parameterizations of Γ (see e.g. [9,18,21]). Recently in [17]), it is proposed to convert the implicit geometry into the graph of an implicitly given height function, leading to a recursive algorithm on the number of dimensions and thus requiring only one-dimensional root-finding and one-dimensional Gaussian quadratures. ...
... 2. Derive an analytical integral formulation I(f, φ) that is easy to discretize, then discretize it (see e.g. [21,22,28,25,27,26,7,8]). We note that this approach computes the integral (1) without using any local parameterizations of Γ 0 . ...
... One choice is to take independent of the level set function φ and the grid. In the work of Smereka [21], the discrete delta-function is concentrated within one grid cell on either side of the interface, and is obtained by discretizing the fundamental solution of the Laplace equation using ghost-points. In the work of Towers [22], the discretized delta function is computed via two different formulations involving the Heaviside function. ...
... The approximation Γ is then used to compute the integral of f over Γ 0 using local parameterizations of Γ (see e.g. [8,14,17]). ...
... One choice is to take independent of the level set function φ and the grid. In the work of Smereka [17], the discrete delta-function is concentrated within one grid cell on either side of the interface, and is obtained by discretizing the fundamental solution of the Laplace equation using ghost-points. In the work of Towers [18], the discretized delta function is computed via two different formulations involving the Heaviside function. ...
... δ is an averaging kernel satisfying some vanishing moment conditions and specifying a tubular neighborhood around Γ 0 , and J(x; d(x)) is the Jacobian that accounts for the change in curvature between nearby level sets and the zero level set Γ 0 . The main advantage and novelty of formulation (3) is that unlike previous techniques in the level set framework (see later Equation (7)) which involve approximating the line or surface integral I 0 using a regularized Dirac-δ function concentrated on Γ 0 [4,5,17,18,21], (3) is equal to I 0 analytically. Errors are therefore due only to the numerical scheme used to discretize (3) instead of both the numerical scheme and the anterior approximation. ...
We provide a new approach for computing integrals over hypersurfaces in the level set framework. In particular, this new approach is able to compute high order approximations of line or surface integrals in the case where the curve or surface has singularities such as corners. The method is based on the discretization (via simple Riemann sums) of the usual line or surface integral formulation used in the level set framework. This integral formulation involves an approximate Dirac delta function supported on a tubular neighborhood around the interface and is an approximation of the line or surface integral one wishes to compute. The novelty of this work is the choice of kernels used to approximate the Dirac delta function. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with singularities, the formulation is not exact but we provide an analytical result relating the severity of the singularity (corner or cusp) with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand has an integrable singularity.
... At the right side, the curvature is correctly shown after multiplying an approximation of a discrete Kronecker delta function, δ K , centered at the interface (φ = 0.5): This is the second addition needed to the proposed equations. It consists of the implementation of a multiplying concentrating interface function [57,62], similar to the delta of Kronecker. Unfortunately, the simulation uses continuous Lagrange functions and thus needs a continuous approximation of the function in Equation (22). ...
... With the curvature, the surface atomic flux is calculated with Equation (7): This is the second addition needed to the proposed equations. It consists of the implementation of a multiplying concentrating interface function [57,62], similar to the delta of Kronecker. Unfortunately, the simulation uses continuous Lagrange functions and thus needs a continuous approximation of the function in Equation (22). ...
The control and prediction of morphological changes in annealed void microstructures is an essential and powerful tool for different semiconductor applications, for example, as part of the production of pressure sensors, resonators, or other silicon structures. In this work, with a focus on the void shape evolution of silicon, a novel simulation approach based on the level-set method is introduced to predict the continuous transformation of initial etched nano/micro-sized cylindrical structures at different annealing conditions. The developed model, which is based on a surface diffusion formulation and built in COMSOL Multiphysics® (Stockholm, Sweden), is introduced and compared to experimental literature data as well as with other analytical approaches. Some advantages of the presented model include the capability of simulating other materials under similar phenomena, the simulation of any possible initial geometry, and the visualization of intermediate steps during the annealing processing.
... However, the phenomenon that is described in this work is a surface one so that a volumetric approach is difficult to define. Furthermore, the equation which drives the surface diffusion would need a fourth order non-linear PDE (Partial Differential Equation) of the phase field variable which is stiff and difficult to solve, especially, at high geometry orders [21]- [23]. There have been some approaches with this method but the computation time was so high that only small variations could be accurately simulated. ...
... It is important to notice that the divergence defined here is the divergence related to the surface and not to the volume because the curvature is a surface phenomenon. In some bibliography, this divergence can be found with the subscript "s" [16], [21], [23]. ...
Simulations of SON structures were carried out by the use of FEM and the arbitrary Lagrangian-Eularian (ALE) method through the software COMSOL Multiphysics®. A novel time-dependent "virtual" curvature algorithm based on non-linear surface diffusion kinetics for FEM simulations is presented. The model describes the evolution from a cylindrical trench etched on silicon to an equilibrium sphere by thermal annealing. An initial aspect ratio (length/diameter) of 2.22-6.66 is determined for creating an ESS. For more than one ESS, the aspect ratio limits are also investigated. With this model, at equilibrium, the step size increases with the length of the initial trench once it is closed while the SON layer increases with the initial cylindrical depth. The temperature enhances the velocity of the evolution.
... We discretize this integral boundary condition (3.3) on each electrode with a first-order quadrature formula based on a first-order discrete dirac function [47]. This discrete Dirac function is by construction positive and non-zero only on the irregular grid points, therefore it can be written as P ∈Ω * h ω P g(P ) + ξ m (P )(U m − u P ) + ε δ m1 U 1 = I m , m = 1, . . . ...
... This discrete Dirac function is by construction positive and non-zero only on the irregular grid points, therefore it can be written as P ∈Ω * h ω P g(P ) + ξ m (P )(U m − u P ) + ε δ m1 U 1 = I m , m = 1, . . . , M. (3.4) with the coefficients ω P that are the weights of the first-order quadrature formula in [47]. The truncature error of this formula is therefore first-order. ...
We propose an immersed boundary scheme for the numerical resolution of the Complete Electrode Model in Electrical Impedance Tomography, that we use as a main ingredient in the resolution of inverse problems in medical imaging. Such method allows to use a Cartesian mesh without accurate discretization of the boundary, which is useful in situations where the boundary is complicated and/or changing. We prove the convergence of our method, and illustrate its efficiency with two dimensional direct and inverse problems.
... This family of unfitted FEM is closely related to classical immersed boundary methods [27,30] and well suited, e.g., for implementation of surface tension effects in two-phase flow models [5,17]. The construction of approximate delta functions for level set methods was addressed in [10,34,37,41]. Müller et al. [28] devised an elegant alternative, which uses the divergence theorem and divergence-free basis functions to reduce integration over an embedded interface to that over a fitted boundary. ...
... The functions H (φ) and δ (φ) represent regularized approximations to H(φ) and the Dirac delta function δ Γ , respectively. The design and analysis of such approximations have received significant attention in the literature during the last two decades [10,18,19,34,37,41]. Many diffuse interface methods [35,36] and level set algorithms [17,31,32] use regularized Heaviside and/or delta functions. ...
We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in embedded domains. We impose the interface conditions weakly and approximate surface integrals by volume integrals. The discretized weak form of the governing equation has the structure of an immersed boundary finite element method. That is, integration is performed over a fixed fictitious domain. Source terms are included to account for interface conditions and extend the boundary data into the complement of the embedded domain. The calculation of these extra terms requires (i) construction of an approximate delta function and (ii) extrapolation of embedded boundary data into quadrature points. We accomplish these tasks using a level set function, which is given analytically or evolved numerically. A globally defined averaged gradient of this approximate signed distance function is used to construct a simple map to the closest point on the interface. The normal and tangential derivatives of the numerical solution at that point are calculated using the interface conditions and/or interpolation on uniform stencils. Similarly to SBM, extrapolation of data back to the quadrature points is performed using Taylor expansions. Computations that require extrapolation are restricted to a narrow band around the interface. Numerical results are presented for elliptic, parabolic, and hyperbolic test problems, which are specifically designed to assess the error caused by the numerical treatment of interface conditions on fixed and moving boundaries in 2D.
... This family of unfitted FEM is closely related to classical immersed boundary methods [28,31] and well suited, e.g., for implementation of surface tension effects in two-phase flow models [6,18]. The construction of approximate delta functions for level set methods was addressed in [11,35,38,42]. Müller et al. [29] devised an elegant alternative, which uses the divergence theorem and divergence-free basis functions to reduce integration over an embedded interface to that over a fitted boundary. ...
... The functions H (φ) and δ (φ) represent regularized approximations to H(φ) and δ Γ , respectively. The design and analysis of such approximations have received significant attention in the literature during the last two decades [11,19,20,35,38,42]. Many diffuse interface methods [36,37] and level set algorithms [18,32,33] use regularized Heaviside and/or delta functions. ...
We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in embedded domains. We impose the interface conditions weakly and approximate surface integrals by volume integrals. The discretized weak form of the governing equation has the structure of an immersed boundary finite element method. A ghost penalty term is included to extend the weak solution into the external subdomain. The calculation of interface forcing terms requires (i) construction of an approximate delta function and (ii) extrapolation of embedded boundary data into quadrature points. We accomplish these tasks using a level set function, which is given analytically or evolved numerically. A globally defined averaged gradient of this approximate signed distance function is used to construct a simple map to the closest point on the interface. The normal and tangential derivatives of the numerical solution at that point are calculated using the interface conditions and/or interpolation on uniform stencils. Similarly to SBM, extrapolation back to the quadrature points is performed using Taylor expansions. The same strategy is used to construct ghost penalty functions and extension velocities. Computations that require extrapolation are restricted to a narrow band around the interface. Numerical results are presented for elliptic, parabolic, and hyperbolic test problems, which are specifically designed to assess the error caused by the numerical treatment of interface conditions on fixed and moving boundaries in 2D.
... vi. Calculate (φ n+1 , c n+1 , ψ n+1 ) according to Eqs. (46), (47), (48) vii. Reinitialize φ if necessary. ...
... Several numerical schemes for approximating δ(φ) and H(φ) have been proposed to address this convergence issue. Engquist proposed using information in the local gradient of the level set function to modify [47], while Smereka borrowed a technique from Green's function theory to discretize the delta function [48]. In our method, the Dirac Delta function δ(φ) and the Heaviside function H(φ) are discretized by the finite difference method developed by Towers in a series of papers [49,50,41,51]. ...
The dynamics of thin, membrane-like structures are ubiquitous in nature. They play especially important roles in cell biology. Cell membranes separate the inside of a cell from the outside, and vesicles compartmentalize proteins into functional microregions, such as the lysosome. Proteins and/or lipid molecules also aggregate and deform membranes to carry out cellular functions. For example, some viral particles can induce the membrane to invaginate and form an endocytic vesicle that pulls the virus into the cell. While the physics of membranes has been extensively studied since the pioneering work of Helfrich in the 1970's, simulating the dynamics of large scale deformations remains challenging, especially for cases where the membrane composition is spatially heterogeneous. Here, we develop a general computational framework to simulate the overdamped dynamics of membranes and vesicles. We start by considering a membrane with an energy that is a generalized functional of the shape invariants and also includes line discontinuities that arise due to phase boundaries. Using this energy, we derive the internal restoring forces and construct a level set-based algorithm that can stably simulate the large-scale dynamics of these generalized membranes, including scenarios that lead to membrane fission. This method is applied to solve for shapes of single-phase vesicles using a range of reduced volumes, reduced area differences, and preferred curvatures. Our results match well the experimentally measured shapes of corresponding vesicles. The method is then applied to explore the dynamics of multiphase vesicles, predicting equilibrium shapes and conditions that lead to fission near phase boundaries.
... The authors state that "taking into account that a higher n sc value produces not only a higher initialization accuracy [...] but also a higher CPU time consumed." Smereka [26] and the series of papers by Wen [31,32,33] are concerned with the numerical evaluation of δfunction integrals in three spatial dimensions. Considering a cuboid intersected by a hypersurface, the concept of Wen is to rewrite the integral over a three-dimensional δ-function as an integral over one of the cell faces, where the integrand is a one-dimensional δ-function. ...
... (25) and plug the result into eq. (16) to obtain the implicit quadratic definition of the boundary curve segment ∂Γ k = Γ∩F k , namely ∂Γ k = {u ∈ S F ,k : u, A k u + u, a k + a k = 0} (26) with the coefficients Table 2 gathers and illustrates the admissible curve classes that emerge from eq. (26). ...
This paper introduces a novel method for the efficient and accurate computation of volume fractions on unstructured polyhedral meshes, where the phase boundary is an orientable hypersurface, implicitly given as the iso-contour of a sufficiently smooth level-set function. Locally, i.e.~in each mesh cell, we compute a principal coordinate system in which the hypersurface can be approximated as the graph of an osculating paraboloid. A recursive application of the \textsc{Gaussian} divergence theorem then allows to analytically transform the volume integrals to curve integrals associated to the polyhedron faces, which can be easily approximated numerically by means of standard \textsc{Gauss-Legendre} quadrature. This face-based formulation enables the applicability to unstructured meshes and considerably simplifies the numerical procedure for applications in three spatial dimensions. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes, showing both high accuracy and third- to fourth-order convergence with spatial resolution.
... which is a second-order approximation [35]. This means that the numerical solution for the initial value problem (Appendix) cannot be more accurate than second-order due to the numerical implementation of the Dirac delta function in the FD scheme. ...
... Fig. 1-c shows values of the ratio R (23) as a function of the Courant number S, ω∆t and k∆x for the Lax-Friedrichs scheme (35). The real part of R-Re(R)-is equal to one for only a single value of the Courant number (S = 1) and the imaginary part of R-Im(R)-is not zero. ...
The Complex-Step-Finite-Difference method (CSFDM) is a very simple methodology that can be implemented in well known numerical techniques helping to improve, for instance in the wave propagation problem, time and/or space derivative based wavefields. We clarify differences between the CSFDM and previous implementations of the Complex-Step (CS) derivative approximation in well known numerical techniques. We study dispersion properties for the one-way and two-way wave equations using the Finite-Difference method (FDM), the Pseudospectral method (PSM), the Finite-Element method (FEM) and the CSFDM, under the influence of a plane wave and Ricker source time functions. We show the gain in numerical accuracy offered by the methodology of the CSFDM over the FDM, PSM and FEM. We finally discuss consequences of the CSFDM in future scenarios and propose directions of study in this area.
... As mentioned before we mainly work with Dirac measure. There are several papers dealing with differential equations with jump as Dirac function, e.g., see [18]. ...
... Remark 3. One can extend the previous results to find a similar discretization of Dirac measure in two dimensions. For more details see [18]. ...
... In order to estimate the analytical sensitivities of the Lagrangian function (15), the discrete form of the gradient has to be found, as well. Due to the Dirac delta term the numerical approximation of the integrals is difficult and several approaches to solve this problem were proposed. ...
... This approach, however, is usually applied to signed distance level-set functions 2 and did not work correctly for the parametrization proposed in this work. This is also the case of other approaches that either could not be applied to a non-signed distance levelset functions [15] or were giving inaccurate results. As a result, an alternative, simple and accurate method was proposed and is described below. ...
Although Topology Optimization is widely used in many industrial applications, it is still in the initial phase of development for highly nonlinear, multimodal and noisy problems, where the analytical sensitivity information is either not available or difficult to obtain. For these problems, including the highly relevant crashworthiness optimization, alternative approaches, relying not solely on the gradient, are necessary. One option are Evolutionary Algorithms, which are well-suited for this type of problems, but with the drawback of considerable computational costs. In this paper we propose a hybrid evolutionary optimization method using a geometric Level-Set Method for an implicit representation of mechanical structures. Hybrid optimization approach integrates gradient information in stochastic search to improve convergence behavior and global search properties. Gradient information can be obtained from structural state as well as approximated via equivalent state or any known heuristics. In order to evaluate the proposed methods, a minimum compliance problem for a standard cantilever beam benchmark case is considered. These results show that the hybridization is very beneficial in terms of convergence speed and performance of the optimized designs.
... The temporal discretization of (19) with g i defined by (25) can be carried out by virtue of standard Runga-Kutta schemes. Note that (25) simply reduces to (15) for uncut cells. ...
... Due to the increasing popularity of the XFEM [24] and related sharp-interface approaches, the numerical integration over implicitly defined sub-domains has received significant attention in recent years. Most approaches presented in literature deliver second order convergence rates in the presence of curved interfaces [3,16,18,[25][26][27][28][29][30][31], and are thus often supplemented by a recursive subdivision strategy in order to retain the formal convergence order of the underlying discretization scheme. ...
We present a higher-order discretization scheme for the compressible Euler and Navier-Stokes equations with immersed boundaries. Our approach makes use of a Discontinuous Galerkin (DG) discretization in a domain that is implicitly defined by means of a level set function. The zero iso-contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary-fitted cells. In order to retain the full order of convergence of the scheme, it is crucial to perform volume and surface integrals in boundary cells with high accuracy. This is achieved using a linear moment-fitting strategy. Moreover, we apply a non-intrusive cell-agglomeration technique that averts problems with very small and ill-shaped cuts. The robustness, accuracy and convergence properties of the scheme are assessed in several two-dimensional test cases for the steady compressible Euler and Navier-Stokes equations. Approximation orders range from zero to four, even though the approach directly generalizes to even higher orders. In all test cases with a sufficiently smooth solution, the experimental order of convergence matches the expected rate for DG schemes. This article is protected by copyright. All rights reserved.
... With ρ being a level set function defined on every grid point in an interested domain Ω, the isosurface Γ induced by ρ is given by Γ = {(x, y, z) ∈ Ω : ρ(x, y, z) = c}, where c is the recommended isovalue. Assume f (x, y, z) is the surface density function defined in Γ, the surface integral of f in Cartesian grids with a uniform mesh can be evaluated by [51,70] ...
Molecular surface representations have been advertised as a great tool to study protein structure and functions, including protein-ligand binding affinity modeling. However, the conventional surface-area-based methods fail to deliver a competitive performance on the energy scoring tasks. The main reason is the lack of crucial physical and chemical interactions encoded in the molecular surface generations. We present novel molecular surface representations embedded in different scales of the element interactive manifolds featuring the dramatically dimensional reduction and accurately physical and biological properties encoders. Those low-dimensional surface-based descriptors are ready to be paired with any advanced machine learning algorithms to explore the essential structure-activity relationships that give rise to the element interactive surface area-based scoring functions (EISA-score). The newly developed EISA-score has outperformed many state-of-the-art models, including various well-established surface-related representations, in standard PDBbind benchmarks.
... Différentes techniques ont été proposées afin d'obtenir de meilleurs ordres de convergence. [96,27,84,98,15,97]. ...
Les simulations d’écoulements diphasiques avec de forts ratios d’échelles, de fortes déformationset des changements de topologie sont complexes et délicates à capturer numériquement.L’utilisation de méthodes numériques robustes et précises est alors primordiale. Un des aspectsimportants est la méthode de suivi d’interface. Ce travail de thèse se concentre sur ce dernierpoint et plus particulièrement sur la méthode Level Set qui est un choix adéquat pour ce typed’applications. Après avoir présenté cette méthode, nous introduisons différentes approches afind’améliorer l’étape de réinitialisation et de réduire le coût en temps de calcul de l’étape advection.Diverses applications liées à l’étude des impacts de gouttes de pluie sur une surface liquide sontaussi présentées afin de valider les méthodes numériques utilisées.
... where δ h is a grid-dependent smoothed Dirac delta function and x i are the grid points. These approaches typically rely on a cancellation of errors in the summation over regularly spaced grid points and it can be subtle to develop convergent schemes [67,18,61,68,81,39], especially higher-order ones [73,74,75]; see also related methods built through application of the co-area formula [17,34,80,70]. ...
A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of h-refinement with q fixed, the quadrature schemes yield an order of accuracy of 2q, where q is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of q-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling q approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of accompanying numerical experiments demonstrates the quadrature algorithm on two- and three-dimensional problems, including randomly generated geometry involving multiple high curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyper-rectangular constraint cells; and boolean operations to compute integrals on overlapping domains.
... Furthermore, the Green's function should be carefully considered. For this, numerical introduction of the Dirac delta function can be of the form δ (x) = 1 ϕ(x/ ) [286][287][288][289][290][291], where one can choose ϕ(x) = 1 2 (1 + cos(πx)) for x 1 and ϕ(x) = 0 for x > 1. In the multi-pulses regime, the accumulated energy in the form of temperature difference is written as follows: ...
This thesis is focused on numerical simulations of the laser interaction with porous materials. A possibility of well-controlled processing is particularly important for the laser based micro-structuring of porous glass and nano-machining of semiconducting porous materials in the presence of metallic nanoparticles. To understand the periodic micro-void structures produced inside porous glass by femtosecond laser pulses, a detailed numerical thermodynamic analysis was performed. The calculation results show the possibility to control laser micro-machining in volume of SiO2. The obtained characteristic dimensions of the structures correlate with the experimental results. Comparing to the porous glass, the mesoporous TiO2 films loaded by Ag ions and nanoparticles support localized plasmon resonances. To identify the optimum parameters of the continuous-wave laser, a multi-physical model considering Ag nanoparticle growth, photo-oxidation, reduction was developed. The performed simulations show that the laser writing speed controls the Ag nanoparticles size. The thermally activated fast growth followed by the photo-oxidation was found to be the main reason for the writing speed-dependent size-change and temperature rises. Writing of mesoporous TiO2 films loaded with Ag nanoparticles by a pulsed laser is, furthermore, promising to provide additional possibilities in the generation of two kinds of nanostructures: laser-induced periodic surface grooves and Ag nanogratingsinside the TiO2 film. To better understand the effects of a pulsed laser, two multi-pulses models are developed to simulate the Ag nanoparticle growth. The obtained results provided new insights into laser micro-processing of porous material and better laser controlling over nanostructuring in porous semiconducting films loaded with metallic nanoparticles.
... Furthermore, the Green's function should be carefully considered. For this, numerical introduction of the Dirac delta function can be of the form δ (x) = 1 ϕ(x/ ) [286][287][288][289][290][291], where one can choose ϕ(x) = 1 2 (1 + cos(πx)) for x 1 and ϕ(x) = 0 for x > 1. In the multi-pulses regime, the accumulated energy in the form of temperature difference is written as follows: ...
Cette thèse porte sur les simulations numériques de l’interaction laser avec des matériaux poreux. Une possibilité de traitement bien contrôlé est particulièrement importante pour la microstructuration laser du verre poreux et le nano-usinage de matériaux poreux semiconducteurs en présence de nanoparticules métalliques. La modélisation auto-cohérente se concentre donc sur une étude détaillée des processus impliqués. En particulier, pour comprendre les structures des micro-vides périodiques produits à l’intérieur du verre poreux par des impulsions laser femtoseconde, une analyse thermodynamique numérique détaillée a été réalisée. Les résultats des calculs montrent la possibilité de contrôler le micro-usinage laser en volume de SiO2 . De plus, les dimensions des structures densifiées par laser sont examinées pour différentes conditions de focalisation à de faibles énergies d’impulsion. Les dimensions caractéristiques obtenues à partir des structures sont corrélées avec les résultats expérimentaux. Comparés au verre poreux, les films mésoporeux TiO2 chargés d’ions Ag et de nanoparticules supportent des ré- sonances plasmoniques localisées. Les films nanocomposites obtenus sont capables de transférer des électrons libres et d’absorber l’énergie laser de manière résonnante, offrant des possibilités supplémentaires pour contrôler la taille des nanoparticules d’Ag. Pour identifier les paramètres optimaux du laser à onde continue, un modèle multi-physique prenant en compte la croissance des nanoparticules d’Ag, photo-oxydation, réduction a été développé. Les simulations réalisées montrent que la vitesse d’écriture laser contrôle la taille des nanoparticules d’Ag. Les calculs ont également représenté une nouvelle vision selon laquelle les nanoparticules d’Ag se développent devant le centre du faisceau laser du fait de la diffusion de chaleur. Il a été démontré que la croissance rapide activée thermiquement suivie d’une photo-oxydation est la principale raison du changement de taille et de température en fonction de la vitesse d’écriture. Un modèle tridimensionnel a été développé et reproduit les lignes écrites au laser. L’écriture de films mésoporeux TiO2 chargés de nanoparticules d’Ag par un laser pulsé promet également d’offrir des possibilités supplémentaires dans la génération de deux types de nanostructures: les rainures de surface périodiques induites par laser (LIPSS) et les nanogratings Ag à l’intérieur du film TiO2 . Pour mieux comprendre les effets d’un laser pulsé, deux modèles multiimpulsions - un semi-analytique et un autre basé sur une méthode par éléments finis (FEM) - sont développés pour simuler la croissance des nanoparticules d’Ag. Le modèle FEM s’avère précis car il traite mieux la diffusion de la chaleur à l’intérieur des films minces TiO2 . Le modèle pourrait être étendu à l’avenir pour comprendre la formation de nanogratings LIPSS et Ag dans de tels milieux en les couplant avec les migrations de nanoparticules, la fusion de surface et l’hydrodynamique.Les résultats obtenus ont ouvert de nouvelles perspectives sur le microtraitement laser des matériaux poreux et un meilleur contrôle laser sur la nanostructuration dans les films semiconducteurs poreux chargés de nanoparticules métalliques.
... Furthermore, the Green's function should be carefully considered. For this, numerical introduction of the Dirac delta function can be of the form δ (x) = 1 ϕ(x/ ) [286][287][288][289][290][291], where one can choose ϕ(x) = 1 2 (1 + cos(πx)) for x 1 and ϕ(x) = 0 for x > 1. In the multi-pulses regime, the accumulated energy in the form of temperature difference is written as follows: ...
This thesis is focused on numerical simulations of the laser interaction with porous materials. A possibility of well-controlled processing is particularly important for the laser based micro-structuring of porous glass and nano-machining of semiconducting porous materials in the presence of metallic nanoparticles. The self-consistent modeling is, therefore, focused on a detailed investigation of the involved processes. Particularly, to understand the periodic micro-void structures produced inside porous glass by femtosecond laser pulses, a detailed numerical thermodynamic analysis was performed. The calculation results show the possibility to control laser micro-machining in volume of SiO2 . Furthermore, the dimensions of laser-densified structures are examined for different focusing conditions at low pulse energies. The obtained characteristic dimensions of the structures correlate with the experimental results. Comparing to the porous glass, the mesoporous TiO2 films loaded by Ag ions and nanoparticles support localized plasmon resonances. The resulted nanocomposite films are capable to transfer free electrons and to resonantly absorb laser energy providing additional possibilities in controlling Ag nanoparticle size.To identify the optimum parameters of the continuous-wave laser, a multi-physical model considering Ag nanoparticle growth, photo-oxidation, reduction was developed. The performed simulations show that the laser writing speed controls the Ag nanoparticles size. The calculations also depicted a novel view that Ag nanoparticles grow ahead of the laser beam center due to the heat diffusion. The thermally activated fast growth followed by the photo-oxidation was found to be the main reason for the writing speed dependent sizechange and temperature rises. A three-dimensional model was developed and reproduced the laser written lines.Writing of mesoporous TiO2 films loaded with Ag nanoparticles by a pulsed laser is, furthermore, promising to provide additional possibilities in the generation of two kinds of nanostructures: laser induced periodic surface grooves (LIPSS) and Ag nanogratingsinside the TiO2 film. To better understand the effects of a pulsed laser, two multi-pulses models - one semi-analytic and another one based on a finite element method (FEM) are developed to simulate the Ag nanoparticle growth. The FEM model is shown to be precise because it better treats heat diffusion inside the TiO2 thin films. The model could be extended in future to understand the formation of LIPSS and Ag nanogratings in such media by coupling with nanoparticle migrations, surface melting and hydrodynamics. The obtained results provided new insights into laser micro-processing of porous material and better laser controlling over nanostructuring in porous semiconducting films loaded with metallic nanoparticles.
... Reference [19] shows that seemingly reasonable methods of approximation may not converge to (4) as the mesh size approaches zero. Since this inconsistency problem came to light there been significant progress on (4) related to finding consistent approximations, obtaining approximations that are better than first order accurate, and constructing proofs of consistency and rates of convergence [1,2,4,10,12,13,18,20,21,23,24,25,27]. It is also the case that seemingly reasonable methods for approximating the integral (1) may not converge to (1) as the discretization parameter approaches zero. ...
This note presents methods for approximating delta functions that are supported on a discrete set of points in R n , n = 2, 3. The delta functions are then used to compute integrals of a type that appear in computational high frequency wave propagation. The concentration points of the delta functions are located at the intersection of the zero level sets of n functions, and thus are not known a priori. The data of the problem are only defined on a regular grid in R n. In [22] we proposed first order and second order accurate finite difference methods for computing these delta functions. In the three-dimensional case we found it necessary to add a gradient normalization preprocessing step in order to guarantee convergence of the integral to the correct value. The novel contribution of this note is a modification to our scheme so that the gradient normalization step is not required, thus simplifying the algorithms and reducing the computational effort. The new algorithms retain their first and second order accuracy. We provide numerical examples to demonstrate convergence in both two and three dimensions.
... One can show that this approach is equivalent to adding penalty terms in finite element methods [3]. Variations in this class involve the choice of regularization applied to the delta functions and the user specified magnitude of the concentrated force [4][5][6]. While these methods have proven useful for low and moderate Reynolds number flows, it is challenging to extend this type of method to higher Reynolds numbers, since the regularization of the delta function deteriorates the local accuracy in resolving the boundary layer. ...
... Subsequent decomposition of the polyhedron into simplices allows for straightforward evaluation of the desired integrals. Smereka [2006] and the series of papers by Wen [2007Wen [ , 2009Wen [ , 2010 are concerned with the numerical evaluation of delta-function integrals in three spatial dimensions. Considering a cuboid intersected by a hypersurface, the concept of Wen is to rewrite the integral over a three-dimensional delta-function as an integral over one of the cell faces, where the integrand is a one-dimensional delta function. ...
This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and thrid- to fourth-order convergence in space.
... The delta function at the boundary is modeled in the discrete mesh as zero everywhere except at the discontinuity where it takes the value of 1/Dx, where Dx is the size of the discretization. 29 The evolution of the reaction-diffusion-convection equation is carried out using a simple Euler method in time with derivatives calculated explicitly. In this case, the vorticity is coupled with the front evolution and it is updated accordingly. ...
Chemical reaction fronts traveling in liquids generate gradients of surface tension leading to fluid motion. This surface tension driven flow, known as Marangoni flow, modifies the shape and the speed of the reaction front. We model the front propagation using the Eikonal relation between curvature and normal speed of the front, resulting in a front evolution equation that couples to the fluid velocity. The sharp discontinuity between the reactants and products leads to a surface tension gradient proportional to a delta function. The Stokes equations with the surface tension gradient as part of the boundary conditions provide the corresponding fluid velocity field. Considering stress free boundaries at the bottom of the liquid layer, we find an analytical solution for the fluid vorticity leading to the velocity field. Solving numerically the appropriate no-slip boundary condition, we gain insights into the role of the boundary condition at the bottom layer. We compare our results with results from two other models for front propagation: the deterministic Kardar-Parisi-Zhang equation and a reaction-diffusion equation with cubic autocatalysis, finding good agreement for small differences in surface tension.
... The computation of the heat release rate requires the integration at a level-set G = 0, where the numerical evaluation utilised the formulation by Smereka (2006). Since the flame tubes are used to track the flame front, the flame tubes are rebuilt with the updated information of the Gfield after the core processes of the simulation have been performed. ...
Numerical simulations aid combustor design to avoid and reduce thermo-acoustic oscillations. Non-linear heat release rate estimation and its modelling are essential for the prediction of saturation amplitudes of limit cycles. The heat release dynamics of flames can be approximated by a Flame Describing Function (FDF). To calculate an FDF, a wide range of forcing amplitudes and frequencies needs to be considered. For this reason, we present a computationally inexpensive level-set approach, which accounts for equivalence ratio perturbations on flames with arbitrarily-complex shapes. The influence of flame parameters and modelling approaches on flame describing functions and time delay coefficient distributions are discussed in detail. The numerically-obtained flame describing functions are compared with experimental data and used in an acoustic network model for limit cycle prediction. A reasonable agreement of the heat release gain and limit cycle frequency is achieved even with a simplistic, analytical velocity fluctuation model. However, the phase decay is over-predicted. For sophisticated flame shapes, only the realistic modelling of large-scale flow structures allows the correct phase decay predictions of the heat release rate response.
... where, δ(G) is the Dirac-delta function and h R (φ) is the heat of reaction. The above integral, equation (2.12), is evaluated numerically using the formulation by Smereka [104]. Note that, for fully premixed flames with constant flame speed, heat release rate oscillations are only due to flame surface area fluctuations induced by velocity perturbations that distort the flame surface. ...
Finding limit cycles and their stability is one of the central problems of nonlinear thermoacoustics. However, a limit cycle is not the only type of self-excited oscillation in a nonlinear system. Nonlinear systems can have quasi-periodic and chaotic oscillations. This thesis examines the different types of oscillation in a numerical model of a ducted premixed flame, the bifurcations that lead to these oscillations and the influence of external forcing on these oscillations. Criteria for the existence and stability of limit cycles in single mode thermoacoustic systems are derived analytically. These criteria, along with the flame describing function, are used to find the types of bifurcation and minimum triggering amplitudes. The choice of model for the velocity perturbation field around the flame is shown to have a strong influence on the types of bifurcation in the system. Therefore, a reduced order model of the velocity perturbation field in a forced laminar premixed flame is obtained from Direct Numerical Simulation. It is shown that the model currently used in the literature precludes subcritical bifurcations and multi-stability. The self-excited thermoacoustic system is simulated in the time domain with many modes in the acoustics and analysed using methods from nonlinear dynamical systems theory. The transitions to the periodic, quasiperiodic and chaotic oscillations are via sub/supercritical Hopf, Neimark-Sacker and period-doubling bifurcations. Routes to chaos are established in this system. It is shown that the single mode system, which gives the same results as a describing function approach, fails to capture the period-2, period-k, quasi-periodic and chaotic oscillations or the bifurcations and multi-stability seen in the multi-modal case, and underpredicts the amplitude. Instantaneous flame images reveal that the wrinkles on the flame surface and pinch off of flame pockets are regular for periodic oscillations, while they are irregular and have multiple time and length scales for quasi-periodic and chaotic oscillations. Cusp formation, their destruction by flame propagation normal to itself, and pinch-off and rapid burning of pockets of reactants are shown to be responsible for generating a heat release rate that is a highly nonlinear function of the velocity perturbations. It is also shown that for a given acoustic model of the duct, many discretization modes are required to capture the rich dynamics and nonlinear feedback between heat release and acoustics seen in experiments. The influence of external harmonic forcing on self-excited periodic, quasi-periodic and chaotic oscillations are examined. The transition to lock-in, the forcing amplitude required for lock-in and the system response at lock-in are characterized. At certain frequencies, even low-amplitude forcing is sufficient to suppress period-1 oscillations to amplitudes that are 90 lower than that of the unforced state. Therefore, open-loop forcing can be an effective strategy for the suppression of thermoacoustic oscillations. This thesis shows that a ducted premixed flame behaves similarly to low-dimensional chaotic systems and that methods from nonlinear dynamical systems theory are superior to the describing function approach in the frequency domain and time domain analysis currently used in nonlinear thermoacoustics.
... where J Irrin is the set of irregular grid points that are inside the SES. Smereka originally proposed a similar scheme [74] for evaluating the surface integration and analyzed its convergence. We have utilized a similar scheme for calculating the surface area of SESs. ...
Solvent excluded surface (SES) is one of the most popular surface definitions in biophysics and molecular biology. In addition to its usage in biomolecular visualization, it has been widely used in implicit solvent models, in which SES is usually immersed in a Cartesian mesh. Therefore, it is important to construct SESs in the Eulerian representation for biophysical modeling and computation. This work describes a software package called Eulerian solvent excluded surface (ESES) for the generation of accurate SESs in Cartesian grids. ESES offers the description of the solvent and solute domains by specifying all the intersection points between the SES and the Cartesian grid lines. Additionally, the interface normal at each intersection point is evaluated. Furthermore, for a given biomolecule, the ESES software not only provides the whole surface area, but also partitions the surface area according to atomic types. Homology theory is utilized to detect topological features, such as loops and cavities, on the complex formed by the SES. The sizes of loops and cavities are measured based on persistent homology with an evolutionary partial differential equation-based filtration. ESES is extensively validated by surface visualization, electrostatic solvation free energy computation, surface area and volume calculations, and loop and cavity detection and their size estimation. We used the Amber PBSA test set in our electrostatic solvation energy, area, and volume validations. Our results are either calibrated by analytical values or compared with those from the MSMS software. © 2017 Wiley Periodicals, Inc.
... We now briefly illustrate the versatility of the jump splice by showing how Proposition 1 can be used to perform integration over implicitly defined surfaces. See [10,69,70] for other approaches to this type of quadrature with level sets. We will use the methods described here to calculate the volume enclosed by an interface when we examine convergence in volume for the Navier-Stokes equations in the next section. ...
We present a general framework for accurately evaluating finite difference operators in the presence of known discontinuities across an interface. Using these techniques, we develop simple-to-implement, second-order accurate methods for elliptic problems with interfacial discontinuities and for the incompressible Navier-Stokes equations with singular forces. To do this, we first establish an expression relating the derivatives being evaluated, the finite difference stencil, and a compact extrapolation of the jump conditions. By representing the interface with a level set function, we show that this extrapolation can be constructed using dimension- and coordinate-independent normal Taylor expansions with arbitrary order of accuracy. Our method is robust to non-smooth geometry, permits the use of symmetric positive-definite solvers for elliptic equations, and also works in 3D with only a change in finite difference stencil. We rigorously establish the convergence properties of the method and present extensive numerical results. In particular, we show that our method is second-order accurate for the incompressible Navier-Stokes equations with surface tension.
... (e) Finally, level-set based approaches [24,72] to deal with boundary cells. ...
With Adaptively Weighted (AW) numerical integration, for a given set of quadrature nodes, order and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain, and then are corrected for the deviation from the original domain using shape sensitivity analysis. In this paper, we demonstrate the application of AW integration scheme in the context of the Finite Cell Method which must perform numerical integration over arbitrary domains without meshing. The standard integration technique employed in FCM is the characteristic function method that converts the continuous integrand over a complex domain into a discontinuous integrand over a simple (box) domain. Then, well known integrand adaptivity techniques are employed to integrate the resulting discontinuous integrand over the box domain. Although this method is simple to implement, it becomes computationally very expensive for realistic complex 3D domains such as sculptures, bones and engines. In contrast, in AW scheme the quadrature weights directly adapt to the complex geometric domain without the need to making the integrand discontinuous leading to superior computational properties. In this paper, we demonstrate the computational efficiency of AW over the characteristic function method as it requires fewer subdivisions and less time to achieve a given accuracy in both two and three dimensions. In addition, AW offers a number of advantages including flexibility in the choice of quadrature points and basis functions.
... The main idea is to reconstruct a piecewise linear approximation of the boundary and compute exactly the boundary integral for a linear approximation of the integrand. There has been many works on improving the computation of surface integrals in the context of the level-set method (Beale 2008;Tornberg et al. 2005;Min and Gibou 2007;Smereka 2006). Here we follow the lead of Gibou (2007, 2008) and we compute boundary integrals as in a finite element method, building a mass matrix. ...
In this paper we introduce a new variant of shape differentiation which is adapted to the deformation of shapes along their normal direction. This is typically the case in the level-set method for shape optimization where the shape evolves with a normal velocity. As all other variants of the original Hadamard method of shape differentiation, our approach yields the same first order derivative. However, the Hessian or second-order derivative is different and somehow simpler since only normal movements are allowed. The applications of this new Hessian formula are twofold. First, it leads to a novel extension method for the normal velocity, used in the Hamilton-Jacobi equation of front propagation. Second, as could be expected, it is at the basis of a Newton optimization algorithm which is conceptually simpler since no tangential displacements have to be considered. Numerical examples are given to illustrate the potentiality of these two applications. The key technical tool for our approach is the method of bicharacteristics for solving Hamilton-Jacobi equations. Our new idea is to differentiate the shape along these bicharacteristics (a system of two ordinary differential equations).
The adjoint method is a popular method used for seismic (full-waveform) inversion today. The method is considered to give more realistic and detailed images of the interior of the Earth by the use of more realistic physics. It relies on the definition of an adjoint wavefield (hence its name) that is the time-reversed synthetics that satisfy the original equations of motion. The physical justification of the nature of the adjoint wavefield is, however, commonly done by brute force with ad hoc assumptions and/or relying on the existence of Green’s functions, the representation theorem and/or the Born approximation. Using variational principles only, and without these mentioned assumptions and/or additional mathematical tools, we show that the time-reversed adjoint wavefield should be defined as a premise that leads to the correct adjoint equations. This allows us to clarify mathematical inconsistencies found in previous seminal works when dealing with viscoelastic attenuation and/or odd-order derivative terms in the equation of motion. We then discuss some methodologies for the numerical implementation of the method in the time domain and to present a variational formulation for the construction of different misfit functions. We here define a new misfit travel-time function that allows us to find consensus for the longstanding debate on the zero sensitivity along the ray path that cross-correlation travel-time measurements show. In fact, we prove that the zero sensitivity along the ray path appears as a consequence of the assumption on the similarity between data and synthetics required to perform cross-correlation travel-time measurements. When no assumption between data and synthetics is preconceived, travel-time Fréchet kernels show an extremum along the ray path as one intuitively would expect.
Molecular surface representations have been advertised as a great tool to study protein structure and functions, including protein-ligand binding affinity modeling. However, the conventional surface-area-based methods fail to deliver a competitive performance on the energy scoring tasks. The main reason is the lack of crucial physical and chemical interactions encoded in the molecular surface generations. We present novel molecular surface representations embedded in different scales of the element interactive manifolds featuring the dramatically dimensional reduction and accurately physical and biological properties encoders. Those low-dimensional surface-based descriptors are ready to be paired with any advanced machine learning algorithms to explore the essential structure-activity relationships that give rise to the element interactive surface area-based scoring functions (EISA-score). The newly developed EISA-score has outperformed many state-of-the-art models, including various well-established surface-related representations, in standard PDBbind benchmarks.
This chapter introduces geometric notations and concepts related to curves and surfaces moving in a flow. It defines level set methods as alternatives to Lagrangian methods for the implicit tracking of these surfaces, with illustrations from classical examples in image processing and fluid mechanics. Finally, it discusses stability issues related to the time discretizations of these methods and proposes a semi-implicit method which is both simple to implement and efficient form the point of view of stability.
A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of h-refinement with q fixed, the quadrature schemes yield an order of accuracy of 2q, where q is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of q-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling q approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of numerical experiments demonstrate the quadrature algorithm on two- and three-dimensional problems, including: randomly generated geometry involving multiple high-curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyperrectangular constraint cells; and boolean operations to compute integrals on overlapping domains.
A new method is proposed for numerically solving the Poisson equation for non-continuous scalar fields on a uniform Cartesian grid. The sharp discontinuity in both the magnitude and the gradient of the scalar field normal to the interface is represented by the numerical solution with second order accuracy at the interface. This is achieved by setting up a composite solution, which is a weighted average of two fictitious scalar fields that together produce the required discontinuity within each interfacial grid cell. A smooth treatment of the Poisson coefficient in a narrow band around the interface allows sharp interfacial jumps to be expressed with second order accuracy on regular grid points around the interface using a standard signed distance function. Moreover, the jump in the gradient tangent to the interface is not needed to enforce the jump in the gradient normal to the interface. The resulting linear system is symmetric and leads to second order accurate solutions on grid points adjacent to the interface. The accuracy of the new framework is compared with other methods.
This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and third- to fourth-order convergence in space.
We present two finite volume schemes to solve a class of Poisson-type equations subject to Robin boundary conditions in irregular domains with piecewise smooth boundaries. The first scheme results in a symmetric linear system and produces second-order accurate numerical solutions with first-order accurate gradients in the L∞-norm (for solutions with two bounded derivatives). The second scheme is nonsymmetric but produces second-order accurate numerical solutions as well as second-order accurate gradients in the L∞-norm (for solutions with three bounded derivatives). Numerical examples are given in two and three spatial dimensions.
We present a discretization method for the multidimensional Dirac distribution. We show its applicability in the context of integration problems, and for discretizing Dirac-distributed source terms in Poisson equations with constant or variable diffusion coefficients. The discretization is cell-based and can thus be applied in a straightforward fashion to Quadtree/Octree grids. The method produces second-order accurate results for integration. Superlinear convergence is observed when it is used to model Dirac-distributed source terms in Poisson equations: the observed order of convergence is 2 or slightly smaller. The method is consistent with the discretization of Dirac delta distribution for codimension one surfaces presented in [1,2]. We present Quadtree/Octree construction procedures to preserve convergence and present various numerical examples, including multi-scale problems that are intractable with uniform grids.
The art gallery and watchman route problems (AGP and WRP) are NP-hard constrained optimization problems concerned with providing static and dynamic sensing, respectively, to environments such that the maximum amount of information is sensed at a minimal cost. What being an NP-hard problem means, practically, is that when an AGP or WRP solution is calculated for a particular time step t, any small change in the environment requires that an entirely new solution must be computed. Extending 3D AGP- and WRP-solving computations into 4D (i.e. considering time's effects on the solutions generated) means that a large number of computational resources would be consumed if the updates to the AGP and WRP solutions are performed serially - since each time step's solution would be computed sequentially. Our particular AGP- and WRP-solving algorithms are built upon the photon mapping algorithm in order to model the information obtainable in the sensed environment. The photon mapping algorithm models the propagation of multispectral photons through an environment and stores the result of the photons' interaction with their environment in a k-d tree data structure called a photon map. Since each virtual photon can operate independently of every other virtual photon, a photon map generated at a particular time step t can be generated independently of every other photon map populated at every other time step using a graphics processing unit (GPU). Thus given an n-sized time sequence, a photon map can be populated by each member of an n-core GPU. Once the photon map is updated, our AGP/WRP-solving algorithms can be executed in parallel over the time sequence using the particular core assigned to a photon map's population. We present the results of our computations and compare both serial- and GPU-based performance.
We present an algorithm providing a heuristic solution to the NP-hard optimization problem known as the watchman route problem (WRP) within a 3D virtual environment testbed populated by simulated unmanned vehicles (UVs). The contribution made by our algorithm is three-fold. First, we utilize photon mapping as our means of representing the information sensed by a UV. Second, we use the photon map to generate an online solution to the closely-related NP-hard art gallery problem (AGP). Third, we use a 3D Chan-Vese segmentation algorithm initialized by our AGP-solver to produce a candidate set of path-planning waypoints. The use of photon mapping with our online AGP solver allows us to adapt UV operation to accommodate variable, less-than-ideal environmental circumstances. The use of our 3D Chan-Vese segmentation algorithm creates a set of candidate waypoints that yield greater visibility coverage when computing the WRP than would be obtainable otherwise. Our algorithm provides for quick learning among the unmanned vehicles operating within the testbed's virtual environment by generating easilytransferrable WRP-solving waypoints. Copyright © 2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.
Understanding how to provide better surveillance in areas not viewable by visible light can arrive by modeling a virtual environment illuminated by photons in the non-visible spectrum and providing the mobile sensing platforms (MSPs) populating these environments with the tools to maximize their sensing capabilities. In order to enhance MSP sensing ability as well as enable MSP route path-planning, we propose a 3D segmentation algorithm based upon the Chan-Vese method to create a connected 3D mesh within the MSPs' photon-mapping-illuminated virtual environment. The resulting segmentation mesh's vertices contain more photons to be sensed by an MSP traversing the mesh than could have been sensed if the MSP had traveled elsewhere. The connectedness of the segmentation mesh gives the MSP uninterrupted travel through these highly-illuminated areas and allows for a variety of mission-planning scenarios. The initialization problem inherent to the Chan-Vese segmentation algorithm is overcome in a novel way by using output from an algorithm solving the art gallery problem to produce an initial segmentation curve comprised of vertices which are highly distinguished from their neighbors. The results of our segmentation algorithm enables an MSP to focus its attention on areas in the 3D environment that maximize the (non-)visible spectrum photons obtainable by their sensors or conversely explore areas have not been well-illuminated.
Numerical methods are studied for the one-dimensional heat equation with a singular forcing term, The delta function is replaced by a discrete approximation and the resulting equation is solved by a Crank–Nicolson method on a uniform grid. The accuracy of this method is analyzed for various choices of . The case where c(t) is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied. These problems serve as a model for the immersed boundary method of Peskin for incompressible flow problems in irregular regions. Some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions.
A level set method for capturing the interface between two fluids is
combined with a variable density projection method to allow for
computation of two-phase flow where the interface can merge/break and
the flow can have a high Reynolds number. A distance function
formulation of the level set method enables one to compute flows with
large density ratios (1000/1) and flows that are surface tension driven;
with no emotional involvement. Recent work has improved the accuracy of
the distance function formulation and the accuracy of the advection
scheme. We compute flows involving air bubbles and water drops, to name
a few. We validate our code against experiments and theory.
We present fast methods for solving Laplace’s and the biharmonic equations on irregular regions with smooth boundaries. The methods used for solving both equations make use of fast Poisson solvers on a rectangular region in which the irregular region is embedded. They also both use an integral equation formulation of the problem where the integral equations are Fredholm integral equations of the second kind. The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region. Fast solvers are then used to compute the extended solution. Aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth.
Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized one-dimensional delta functions. It is independent of the irregularity relative to the grid. In the second method, the regularization is constructed from a one-dimensional regularization that is extended to multi-dimensions with a variable support depending on the orientation of the singularity relative to the computational grid. Convergence analysis and numerical results are given.
Interfaces have a variety of boundary conditions (or jump conditions) that need to be enforced. The Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid Euler equations and has been extended to treat more general discontinuities such as shocks, detonations, and deflagrations and compressible viscous flows. In this paper, a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient Poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous. This new method is robust and easy to implement even in three spatial dimensions. Furthermore, the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient Poisson equation in the absence of interfaces allowing for straightforward application of standard “black box” solvers.
We develop a fast method to localize the level set method of Osher and Sethian (1988, J. Comput. Phys.79, 12) and address two important issues that are intrinsic to the level set method: (a) how to extend a quantity that is given only on the interface to a neighborhood of the interface; (b) how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface. This fast local level set method reduces the computational effort by one order of magnitude, works in as much generality as the original one, and is conceptually simple and easy to implement. Our approach differs from previous related works in that we extract all the information needed from the level set function (or functions in multiphase flow) and do not need to find explicitly the location of the interface in the space domain. The complexity of our method to do tasks such as extension and distance reinitialization is O(N), where N is the number of points in space, not O(N log N) as in works by Sethian (1996, Proc. Nat. Acad. Sci. 93, 1591) and Helmsen and co-workers (1996, SPIE Microlithography IX, p. 253). This complexity estimation is also valid for quite general geometrically based front motion for our localized method.
We present a new method for the rapid, high order accurate evaluation of certain volume integrals in potential theory on general irregular regions. The kernels of the integrals are either a fundamental solution, or a linear combination of the derivatives of a fundamental solution of a second-order linear elliptic differential equation. Instead of using a standard quadrature formula or the exact evaluation of any integral, the methods rely on rapid methods of solving the differential equation which the kernel is the solution of. Therefore, the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid, and the method requires no evaluation of the kernel.
Singular terms in differential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi-dimensions. New high order multi-dimensional techniques for differential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions in level-set methods to extend one dimensional regularization to higher dimensions may produce O(1) errors.
this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coefficients or/and singular sources along the interfaces. The moving interfaces then are updated using the newly developed fast level set formulation which involves computation only inside some small tubes containing the interfaces. This method combines the advantage of the two approaches and gives a second-order Eulerian discretization for interface problems. Several key steps in the implementation are addressed in detail. This newapproach is then applied to Hele-Shaw flow, an unstable flow involving two fluids with very different viscosity. 1997 Academic Press L
. A second order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in a previous paper (SIAM J. Numer. Anal., 31(1994), pp. 1019--1044). The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used. Key words. Stokes flow, creeping flow, interface tracking, discontinuous coefficients, immersed interface methods, Cartesian grids, bubbles. AMS subject clas...
A front-tracking method for the computation of multiphase flow
- Unverdi