Nodal Basis Functions Research Articles

Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction

Abstract A finite-element method for computing the electric field in a 3-D conductivity model of the Earth for plane wave sources, thus enabling magnetotelluric responses to be calculated, is presented. The method incorporates in the iterative solution of the electric-field system of equations the divergence correction technique introduced for finite-difference solutions by Smith (1996). The correction technique accelerates the development of the discontinuity of the normal component of the approximate electric field across conductivity discontinuities. The convergence rate of the iterative solution is improved significantly, especially for low frequencies. The correction technique involves computing the divergence of the current density for the approximate electric field, computing the static potential whose source is this divergence of the current density, and ‘correcting’ the approximate electric field by subtracting from it the gradient of the potential. This is repeated at regular intervals during the iterative solution of the electric-field system of equations. For the method presented here, the Earth model is discretised using a rectilinear mesh comprising uniform cells. Edge-element basis functions are used to approximate the electric field and nodal basis functions are used to approximate the correction potential. The Galerkin method is used to derive the systems of equations for the approximate electric field and correction potential from the respective differential equations. A bi-conjugate gradient solver was found to be adequate for the system of equations for the correction potential; a generalised minimum residual solver was found to be better for the electric-field system of equations. The method is illustrated using the COMMEMI 3D-1A and 3D-2A models.

Hp Adaptive finite elements based on derivative recovery and superconvergence

In this paper, we present a new approach to hp-adaptive finite element methods. Our a posteriori error estimates and hp-refinement indicator are inspired by the work on gradient/derivative recovery of Bank and Xu (SIAM J Numer Anal 41:2294–2312, 2003; SIAM J Numer Anal 41:2313–2332, 2003). For element τ of degree p, R(∂puhp), the (piece-wise linear) recovered function of ∂pu is used to approximate \({|\varepsilon|_{1,\tau} = |\hat{u}_{p+1} - u_{p}|_{1,\tau}}\) , which serves as our local error indicator. Under sufficient conditions on the smoothness of u, it can be shown that \({\|{\partial^{p}(\hat{u}_{p+1} - u_{p})\|_{0,\Omega}}}\) is a superconvergent approximation of \({\|(I - R){\partial}^p u_{hp}\|_{0,\Omega}}\) . Based on this, we develop a heuristic hp-refinement indicator based on the ratio between the two quantities on each element. Also in this work, we introduce nodal basis functions for special elements where the polynomial degree along edges is allowed to be different from the overall element degree. Several numerical examples are provided to show the effectiveness of our approach.

A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented.

A Comparison of Higher Order Nodal- and Edge-Basis Functions in the MFIE on Rational BÉzier Geometries

Higher order nodal basis functions for representing equivalent surface currents on antennas and scatterers are introduced. The performance of the nodal basis is evaluated by comparing two existing higher order edge bases, using the magnetic field integral equation (MFIE) formulation for scattering by a perfect electric conductor (PEC) sphere and icosahedron as test problems. Both nodal and edge bases are implemented on rational Bezier patches, giving an exact representation of the surfaces, free from geometrical error. The accuracy of the numerical solutions obtained with the three different bases for both the surface current and the radar cross section (RCS) are compared, and it is shown that in general the nodal bases give better accuracy than the edge bases for equal computational cost.

Polynomial basis functions on pyramidal elements

AbstractPyramidal elements are necessary to effect the transition from tetrahedral to hexahedral elements, a common requirement in practical finite element applications. However, existing pyramidal transition elements suffer from degeneracy or other numerical difficulties, requiring, at the least, warnings and care in their use. This paper presents a general technique for the construction of nodal basis functions on pyramidal finite elements. General forms for basis functions of arbitrary order are presented. The basis functions so derived are fully conformal and free of degeneracy. Copyright © 2007 John Wiley & Sons, Ltd.

Finite elements on evolving surfaces

In this article, we define a new evolving surface finite-element method for numerically approximating partial differential equations on hypersurfaces (t) in n+1 which evolve with time. The key idea is based on approximating (t) by an evolving interpolated polyhedral (polygonal if n = 1) surface h(t) consisting of a union of simplices (triangles for n = 2) whose vertices lie on (t). A finite-element space of functions is then defined by taking the set of all continuous functions on h(t) which are linear affine on each simplex. The finite-element nodal basis functions enjoy a transport property which simplifies the computation. We formulate a conservation law for a scalar quantity on (t) and, in the case of a diffusive flux, derive a transport and diffusion equation which takes into account the tangential velocity and the local stretching of the surface. Using surface gradients to define weak forms of elliptic operators naturally generates weak formulations of elliptic and parabolic equations on (t). Our finite-element method is applied to the weak form of the conservation equation. The computations of the mass and element stiffness matrices are simple and straightforward. Error bounds are derived in the case of semi-discretization in space. Numerical experiments are described which indicate the order of convergence and also the power of the method. We describe how this framework may be employed in applications. Key Words: finite elements; evolving surfaces; conservation; diffusion; existence; error estimates; computations

Biorthogonal bases with local support and approximation properties

We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p - 1. Working with Gaus-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.

Nodal DG-FEM solution of high-order Boussinesq-type equations

A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.

The Morley element for fourth order elliptic equations in any dimensions

In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general n-dimensional Morley element consists of all quadratic polynomials defined on each n-simplex with degrees of freedom given by the integral average of the normal derivative on each (n-1)-subsimplex and the integral average of the function value on each (n-2)-subsimplex. Explicit expressions of nodal basis functions are also obtained for this element on general n-simplicial grids. Convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation.

Aggregation-Based Algebraic Multilevel Preconditioning

We propose a preconditioning technique that is applicable in a "black box" fashion to linear systems arising from second order scalar elliptic PDEs discretized by finite differences or finite elements with nodal basis functions. This technique is based on an algebraic multilevel scheme with coarsening by aggregation. We introduce a new aggregation method which, for the targeted class of applications, produces semicoarsening effects whenever desirable, while the number of nodes is decreased by a factor of about 4 at each level, regardless of the problem at hand. Moreover, the number of nonzero entries per row in the successive coarse grid matrices remains approximately constant, ensuring small set up cost and modest memory requirements. This aggregation technique can be used in an algebraic multigrid (AMG)-like framework, but better results are obtained with an algebraic multilevel scheme based on a block approximate factorization of the matrix. In this scheme, the block pivot corresponding to fine grid nodes is approximated by a modified incomplete LU (MILU) factorization. To enhance robustness and avoid any potential breakdown, the coarsening process is refined by recasting as "coarse" fine grid nodes for which the corresponding pivot in this MILU factorization would be negative or too small. Numerical results display the efficiency, the scalability, and the robustness of the resulting preconditioner on a wide set of discrete scalar PDE problems, ranging from the two-dimensional Poisson equation to three-dimensional convection-diffusion problems with high Reynolds number and strongly varying convection.

Anisotropic Mesh Refinement: The Conditioning of Galerkin Boundary Element Matrices and Simple Preconditioners

In this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element Galerkin approximation (using nodal basis functions) of a bounded, symmetric bilinear form which is elliptic on a Sobolev space of real index m \in [-1,1]. The key point is that the finite element mesh is required to be neither quasi-uniform nor shape-regular, so that our theory allows anisotropic meshes often used in practice. (However, we assume that the polynomial degree of the elements is fixed.) Our bounds indicate the ill-conditioning which can arise from anisotropic mesh refinement. In addition we obtain spectral bounds for the diagonally scaled stiffness matrix, which indicate the improvement provided by this simple preconditioning. For the special case of boundary integral operators on a two-dimensional screen in R3, numerical experiments show that our bounds are sharp. We find that diagonal scaling essentially removes the ill-conditioning due to mesh degeneracy, leading to the same asymptotic growth in the condition number as arises for a quasi-uniform mesh refinement. Our results thus generalize earlier work by Bank and Scott [SIAM J. Numer. Anal., 26 (1989), pp. 1383-1394] and Ainsworth, McLean, and Tran [SIAM J. Numer. Anal., 36 (1999), pp. 1901-1932] for the shape-regular case.

DGTD methods using modal basis functions and symplectic local time-stepping

The discontinuous Galerkin time domain (DGTD) methods are now widely used for the solution of wave propagation problems. Able to deal with unstructured meshes past complex geometries, they remain fully explicit with easy parallelization and extension to high orders of accuracy. Still, modal or nodal local basis functions have to be chosen carefully to obtain actual numerical accuracy. Concerning time discretization, explicit non-dissipative energypreserving time-schemes exist, but their stability limit remains linked to the smallest element size in the mesh. Symplectic algorithms, based on local-time stepping or local implicit scheme formulations, can lead to dramatic reductions of computational time, which is shown here on two-dimensional acoustics problems.

Quadratic immersed finite element spaces and their approximation capabilities

This paper discusses a class of quadratic immersed finite element (IFE) spaces developed for solving second order elliptic interface problems. Unlike the linear IFE basis functions, the quadratic IFE local nodal basis functions cannot be uniquely defined by nodal values and interface jump conditions. Three types of one dimensional quadratic IFE basis functions are presented together with their extensions for forming the two dimensional IFE spaces based on rectangular partitions. Approximation capabilities of these IFE spaces are discussed. Finite element solutions based on these IFE for representative interface problems are presented to further illustrate capabilities of these IFE spaces.

Analysis of Defect in Extreme UV Lithography Mask Using a Modal Method Based on Nodal B-Spline Expansion

This paper details to an electromagnetic modeling of an extreme ultraviolet (EUV) lithography mask. For that purpose, a modal method based on a spline nodal expansion (MMSNE) is presented. The results obtained using first, and second-order splines as basis functions are compared with those obtained using other modal methods, such as modal method by Fourier expansion (MMFE). The agreement between the results obtained using different methods is very good, and a convergence test is also performed. The spline nodal basis function implemented in this paper is the first step toward the realization of a multiresolution scheme that is expected to perform much more efficiently than conventional schemes.

Electromagnetic Field Analysis of Rectangular Superconductor With Large Aspect Ratio in Arbitrary Orientated Magnetic Fields

An electromagnetic field analysis of high temperature superconductors with large cross-sectional aspect ratios was carried out. This analysis used a numerical model employing FEM, formulated with current vector potential and magnetic scalar potential. The model conductor with a large cross-sectional aspect ratio simulates a YBCO coated conductor. Edge basis functions and a nodal basis function were used for vector potentials and a scalar potential, respectively. The AC loss was calculated from the temporal evolution of the current distribution. AC transport current was supplied and an AC transverse magnetic field with various orientations was applied to the model conductor to calculate the AC losses at various conditions. The numerically calculated AC losses were compared with the analytical values and numerical values.

Algorithm 839

Much of finite element computation is constrained by the difficulty of evaluating high-order nodal basis functions. While most codes rely on explicit formulae for these basis functions, we present a new approach that allows us to construct a general class of finite element basis functions from orthonormal polynomials and evaluate and differentiate them at any points. This approach relies on fundamental ideas from linear algebra and is implemented in Python using several object-oriented and functional programming techniques.

A variable order wavelet method for the sparse representation of layer potentials in the non-standard form

We discuss a variable order wavelet method for boundary integral formulations of elliptic boundary value problems. The wavelet basis functions are transformations of standard nodal basis functions and have a variable number of vanishing moments. For integral equations of the second kind we will show that the non-standard form can be compressed to contain only O(N) non-vanishing entries while retaining the asymptotic converge of the full Galerkin scheme, where N is the number of degrees of freedom in the discretization.

A variable order wavelet method for the sparse representation of layer potentials in the non-standard form

We discuss a variable order wavelet method for boundary integral formulations of elliptic boundary value problems. The wavelet basis functions are transformations of standard nodal basis functions and have a variable number of vanishing moments. For integral equations of the second kind we will show that the non-standard form can be compressed to contain only O ( N ) non-vanishing entries while retaining the asymptotic converge of the full Galerkin scheme, where N is the number of degrees of freedom in the discretization.

An hp/Spectral Element Model for Efficient Long-Time Integration of Boussinesq-Type Equations

We present an hp/spectral element method for modelling one-dimensional nonlinear dispersive water waves, described by a set of enhanced Boussinesq-type equations. The model uses nodal basis functions of arbitrary order in space and the third-order Adams-Bashforth scheme to advance in time. Numerical computations are used to show that the hp/spectral element model exhibits exponential convergence. The model is compared to two numerical methods frequently used for solving Boussinesq-type equations; a finite element model using linear basis functions and a finite difference model using a five-point stencil for estimating the first-order derivatives. Using numerical examples, we show that the hp/spectral element model gives great savings in computational time, compared to the other models, if: (i) highly accurate results Eire requested, or, more importantly, (ii) results of “engineering accuracy” are called for in combination with long-time integration.

A volume/surface potential formulation of the method of moments applied to electromagnetic scattering

A new integral method to compute the electromagnetic scattering by general inhomogeneous dielectric bodies is presented. The method is based on a volume/surface equivalence principle and it uses the electromagnetic potentials as primary unknowns. The main advantage of this method comes from the continuity of the potentials across interfaces separating different media so that very easy to handle nodal basis functions can be used to solve the integral equations with the method of moments.