Local Mesh Refinement Strategy Research Articles

Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines

Abstract We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation and adaptivity procedure are applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated into a scan-based analysis workflow, capable of generating error-controlled results from scan data without the need for extensive user interactions or interventions.

On transmission Zeros of piezoelectric structures

The evaluation of transmission zeros is of great importance for the control engineering applications. The structures equipped with piezoelectric patches are complex to model and usually require finite element approaches supplemented by model reduction. This study rigorously investigates the influence of mesh size, model reduction, boundary conditions (free and clamped), and sensor/actuator configuration (collocated and non-collocated) on the evaluation of transmission zeros of the piezoelectric structures. The numerical illustrations are presented for a thin rectangular plate equipped with a single pair of piezoelectric voltage sensor/ voltage actuator. Through the examples considered in this study, a link is presented between the static response (or static deflected shape) and the transmission zeros of the piezoelectric structures. This interesting observation forms the basis of: (i) a local mesh refinement strategy for computationally efficient estimation of the transmission zeros and (ii) a physical interpretation of the pole-zero pattern in the case of piezoelectric structures. The physical interpretation developed in this study helps in qualitatively explaining the pole-zero patterns observed for different configurations. It is also shown that this understanding of the relation between the static deformed shape and the transmission zeros can be used by the practitioners to modify the pole-zero pattern through a careful selection of the orientation and the size of the piezoelectric patches.

The effects of transaction costs and illiquidity on the prices of volatility derivatives

The prices of assets differ when considering transaction costs and their illiquidity, thus affecting their returns. As volatility derivatives such as variance swaps, gamma swaps, corridor variance swaps and volatility swaps depend on the expected values of these returns, we develop a high-order algorithm so that differences in the fair values of the volatility derivatives under different transaction costs and illiquidity models can be precisely observed. A high-order finite-difference algorithm is used to solve a framework of partial differential equations. A local mesh refinement strategy is applied at the singularities that appear in the pricing problem of some volatility derivatives so that nonuniform five-point stencil finite differences can be implemented in space. The time-stepping is dealt with by an efficient iterative scheme coupled with a Richardson extrapolation and Carathéodory–Fejér approximations. Consequently, the highly convergent solutions are used to analyze the effect of transaction costs and illiquidity on the fair values of volatility derivatives, and the results demonstrate an increase in these values when different transaction costs and liquidity models are implemented.

Adaptive analysis using scaled boundary finite element method in 3D

In this paper, an adaptive refinement technique using the scaled boundary finite element method (SBFEM) is proposed. The salient feature of this technique is that it is not required to regenerate the mesh for the whole model during the iterations. To this end, a local mesh refinement strategy is implemented based on a polytree algorithm in three dimensions, which can be applied to polyhedral elements with arbitrary number of nodes, edges and faces. These elements constructed by the SBFEM can be used in analysis with their boundaries discretized only, which reduce the difficulty to connect elements with different sizes. An explicit residual based error indicator is developed using the discontinuity of the stress field to guide the adaptive mesh refinement. The accuracy and efficiency of the proposed method are demonstrated using five numerical examples, including complex geometry and stress singularity.

Adaptive Finite Element Method of Lines with Local Mesh Refinement in Maximum Norm Based on Element Energy Projection Method

The reliable and efficient self-adaptive analysis is a modern goal of various numerical computations. Most adaptivity methods, however, adopt energy norm to measure errors, which may not be the most natural and convenient means, e.g., for problems with locally singular gradient of displacement. Based on the Element Energy Projection (EEP) super-convergent technique in the Finite Element Method of Lines (FEMOL) which is a general and powerful semi-discrete method, reliable error estimates of displacements in maximum norm can be obtained anywhere on the FEMOL mesh and hence adaptive FEMOL by maximum norm becomes feasible. However, to tackle singularity problems effectively and efficiently, an automatic and flexible local mesh refinement strategy is required to generate meshes of high quality for more efficient adaptive FEMOL analysis. Taking the two-dimensional Poisson equation as the model problem, the paper firstly introduces the FEMOL and EEP methods with interface sides resulting from local mesh refinement. Then a local mesh refinement strategy and corresponding adaptive algorithm are presented. The numerical results given show that the proposed adaptive FEMOL with local mesh refinement can produce displacement solutions satisfying the specified tolerances in maximum norm and the adaptively-generated meshes reasonably reflect the local difficulties inherent in the physical problems without much redundant accuracy.

An efficient variable-node XFEM for modeling multiple crack growth: A Matlab object-oriented implementation

We present a Matlab object-oriented implementation of an efficient variable-node extended finite element method (XFEM) for multiple crack growth simulation in brittle materials. To enhance the accuracy of the present method, not only the local mesh refinement strategy is used, but also the variable-node elements are adopted to directly link different scale elements, saving the computational cost. In order to efficiently conduct coalescence detection, a modification is also presented. A stability analysis is performed to determine active crack tips, and the crack length control scheme is employed to model multiple crack growth. Mesh is locally refined simultaneously with the extension of cracked area via a straightforward manner. A general computer program structure for the Matlab object-oriented programming code for the variable-node XFEM is described. All those desirable features of the present work are demonstrated through five numerical examples with branching in linear-elastic fracture mechanics. In addition, this work provides an efficient computer code written by Matlab programming language for computational fracture simulation, which is for educational and academic uses.

Constructing 3D Self-Supporting Surfaces with Isotropic Stress Using 4D Minimal Hypersurfaces of Revolution

This article presents a new computational framework for constructing 3D self-supporting surfaces with isotropic stress. Inspired by the self-supporting property of catenary and the fact that catenoid (the surface of revolution of the catenary curve) is a minimal surface, we discover the relation between 3D self-supporting surfaces and 4D minimal hypersurfaces (which are 3-manifolds). Lifting the problem into 4D allows us to convert gravitational forces into tensions and reformulate the equilibrium problem to total potential energy minimization, which can be solved using a variational method. We prove that the hyper-generatrix of a 4D minimal hyper-surface of revolution is a 3D self-supporting surface, implying that constructing a 3D self-supporting surface is equivalent to volume minimization. We show that the energy functional is simply the surface’s gravitational potential energy, which in turn can be converted into a surface reconstruction problem with mean curvature constraint. Armed with our theoretical findings, we develop an iterative algorithm to construct 3D self-supporting surfaces from triangle meshes. Our method guarantees convergence and can produce near-regular triangle meshes, thanks to a local mesh refinement strategy similar to centroidal Voronoi tessellation. It also allows users to tune the geometry via specifying either the zero potential surface or its desired volume. We also develop a finite element method to verify the equilibrium condition on 3D triangle meshes. The existing thrust network analysis methods discretize both geometry and material by approximating the continuous stress field through uniaxial singular stresses, making them an ideal tool for analysis and design of beam structures. In contrast, our method works on piecewise linear surfaces with continuous material. Moreover, our method does not require the 3D-to-2D projection, therefore it also works for both height and non-height fields.

A high-order RBF-FD method for option pricing under regime-switching stochastic volatility models with jumps

In this paper, we develop a high-order radial basis function finite difference (RBF-FD) approximation on a five-point stencil for pricing options under the regime-switching stochastic volatility models with log-normal and contemporaneous jumps (SVCJ). We fully exploit the capabilities of the RBF-FD to perform the interpolation, differentiation and integration approximations tasks required in the numerical solution of the SVCJ pricing partial integro-differential equation (PIDE). The resulting systems of equations are sparse and our treatment of the non-local integro discretisation allows an efficient implementation based on the fast Fourier transform (FFT) algorithm. We show that a local mesh refinement strategy gives high-order convergent solutions for both the European and the path dependent barrier option prices.

An IMEX‐BDF2 compact scheme for pricing options under regime‐switching jump‐diffusion models

In this paper, an implicit‐explicit two‐step backward differentiation formula (IMEX‐BDF2) together with finite difference compact scheme is developed for the numerical pricing of European and American options whose asset price dynamics follow the regime‐switching jump‐diffusion process. It is shown that IMEX‐BDF2 method for solving this system of coupled partial integro‐differential equations is stable with the second‐order accuracy in time. On the basis of IMEX‐BDF2 time semi‐discrete method, we derive a fourth‐order compact (FOC) finite difference scheme for spatial discretization. Since the payoff function of the option at the strike price is not differentiable, the results show only second‐order accuracy in space. To remedy this, a local mesh refinement strategy is used near the strike price so that the accuracy achieves fourth order. Numerical results illustrate the effectiveness of the proposed method for European and American options under regime‐switching jump‐diffusion models.

On the Variable Two-Step IMEX BDF Method for Parabolic Integro-differential Equations with Nonsmooth Initial Data Arising in Finance

In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving parabolic partial integro-differential equations (PIDEs), which describe the jump-diffusion option pricing model in finance. It is shown that the variable step-size IMEX BDF2 method is stable for abstract PIDEs under suitable time step restrictions. Based on the time regularity analysis of abstract PIDEs, the consistency error and the global error bounds for the variable step-size IMEX BDF2 method are provided. After time semidiscretization, spatial differential operators are treated by using finite difference methods, and the jump integral is computed using the composite trapezoidal rule. A local mesh refinement strategy is also considered near the strike price because of the nonsmoothness of the payoff function. Numerical results illustrate the effectiveness of the proposed method for European and American options under jump-diffusion models.

Generation of surface plasmon-polaritons by edge effects

By using numerical and analytical methods, we describe the generation of fine-scale lateral electromagnetic waves, called surface plasmon-polaritons (SPPs), on atomically thick, metamaterial conducting sheets in two spatial dimensions (2D). Our computations capture the two-scale character of the total field and reveal how each edge of the sheet acts as a source of an SPP that may dominate the diffracted field. We use the finite element method to numerically implement a variational formulation for a weak discontinuity of the tangential magnetic field across a hypersurface. An adaptive, local mesh refinement strategy based on a posteriori error estimators is applied to resolve the pronounced two-scale character of wave propagation and radiation over the metamaterial sheet. We demonstrate by numerical examples how a singular geometry, e.g., sheets with sharp edges, and sharp spatial changes in the associated surface conductivity may significantly influence surface plasmons in nanophotonics.

RBF-FD schemes for option valuation under models with price-dependent and stochastic volatility

Radial basis functions based finite difference schemes for the solution of partial differential equations have the advantage that an optimal choice of the shape parameter can yield better accuracies than standard finite difference discretisations based on the same number of nodal points. Such schemes known as local radial basis functions methods are considered for the pricing of options under the constant elasticity of variance and the Heston stochastic volatility model. A general methodology for approximating first and second order derivative terms in the finance pdes is presented and the resulting schemes are applied for option valuation. For one-dimensional problems, we derive a compact-RBF scheme which achieves a higher order accuracy when combined with a local mesh refinement strategy. Numerical results and comparisons made for European, American and barrier options illustrate the good performances of the localized radial basis functions methods.

A high-order finite difference method for option valuation

In this paper, we propose the use of an efficient high-order finite difference algorithm to price options under several pricing models including the Black–Scholes model, the Merton’s jump–diffusion model, the Heston’s stochastic volatility model and the nonlinear transaction costs or illiquidity models. We apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non-uniform high-order five-point stencil finite difference scheme. For linear models, the time-stepping is dealt with by using an exponential time integration scheme with Carathéodory–Fejér approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices. Nonlinear Black–Scholes equations are solved using an efficient iterative scheme coupled with a Richardson extrapolation. Our numerical experiments clearly demonstrate the high-order accuracy of the proposed finite difference method, making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems.

3D finite element modeling of marine controlled-source electromagnetic fields using locally refined unstructured meshes

We present a 3D forward modeling algorithm of marine controlled-source electromagnetic (CSEM) fields using finite element method based on unstructured meshes. As the unstructured mesh is incorporated, the algorithm can easily accommodate complex structures, such as dipping interfaces and rough topography. Secondary coupled-potential formulation of Maxwell's equations allows to avoid the singularities introduced by the sources. The primary electromagnetic fields excited by electric dipole source in layered isotropic media are derived from the semi-analytical formulas based on Schelkunoff potentials. To improve the accuracy and efficiency of the algorithm, we propose an effective local mesh refinement strategy, where the elements near the receiver locations and in the abnormal areas are refined using the volume constraints. We validate the finite element code against a 1D and 2D reservoir model and study the performance of the local mesh refinement strategy. The Canonical disc model illustrates the utility of the finite element method for 3D CSEM modeling. The bathymetry model study shows that the seafloor topography gives an important response which needs to be reproduced by numerical modeling to avoid the misinterpretation of measurements.

Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H1(Ω)‐norm, L2(Ω)‐norm and L2(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.

On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H 1-Stability of L 2-Projection

Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations that consist of simplices. For the 2D case, we prove that the mesh-closure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L 2-projection onto lowest-order Courant finite elements (P1-FEM) is always H 1-stable. Throughout, no additional assumptions on the initial triangulation are imposed. Our analysis thus improves results of Binev et al. (Numer. Math. 97(2):219–268, 2004), Carstensen (Constr. Approx. 20(4):549–564, 2004), and Stevenson (Math. Comput. 77(261):227–241, 2008) in the sense that all assumptions of their theorems are removed. Consequently, our results relax the requirements under which adaptive finite element schemes can be mathematically guaranteed to convergence with quasi-optimal rates.

Fourth‐order compact scheme with local mesh refinement for option pricing in jump‐diffusion model

AbstractThe value of a contingent claim under a jump‐diffusion process satisfies a partial integro‐differential equation. A fourth‐order compact finite difference scheme is applied to discretize the spatial variable of this equation. It is discretized in time by an implicit‐explicit method. Meanwhile, a local mesh refinement strategy is used for handling the nonsmooth payoff condition. Moreover, the numerical quadrature method is exploited to evaluate the jump integral term. It guarantees a Toeplitz‐like structure of the integral operator such that a fast algorithm is feasible. Numerical results show that this approach gives fourth‐order accuracy in space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011

High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems

We derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two-dimensional convection diffusion equation. A multilevel local mesh refinement strategy is used to deal with the local singularity problem. A corresponding multilevel multigrid method is designed to solve the resulting sparse linear system. Numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution. Our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems.

An Adaptive Finite Element Method for Two-Phase Stefan Problems in Two Space Dimensions. II: Implementation and Numerical Experiments

An adaptive local mesh refinement strategy for two-phase Stefan problems is discussed in light of its efficiency and computational complexity. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature and a fourth one, in the event of mushy regions, to equidistribute $L^1 $-interpolation errors for enthalpy within the mush. If certain quality mesh tests fail, then the current mesh is discarded and a new one completely regenerated by an efficient mesh generator, which in turn is briefly described. A typical triangulation is strongly graded to become very fine near computed interfaces and coarse away from them. Consecutive meshes are not compatible. The use of quadtree data structures is discussed as a means to reach a nearly optimal computational complexity in various tasks to be performed, mainly in generating a mesh and interpolating. Various implementation details are given so as to derive the computational complexity of each relevant subroutine. The approximation of both solutions and interfaces is drastically improved. The proposed method is robust in that it can handle the formation of cusps and mushy regions as well as the spontaneous appearance of phases. It is also superior in terms of computing time or a desired accuracy. Several numerical experiments illustrate these facts and provide quantitative information about each task complexity.