Quadratic Basis Functions Research Articles

A Finite Subelement Generalization of the Variational Nodal Method

The variational nodal method is generalized by dividing each spatial node into a number of triangular finite elements designated as subelements. The finite subelement trial functions allow for explicit geometry representations within each node, thus eliminating the need for nodal homogenization. The method is implemented within the Argonne National Laboratory code VARIANT and applied to two-dimensional multigroup problems.Eigenvalue and pin-power results are presented for a four-assembly Organization for Economic Cooperation and Development/Nuclear Energy Agency benchmark problem containing enriched UO2 and mixed oxide fuel pins. Our seven-group model combines spherical or simplified spherical harmonic approximations in angle with isoparametric linear or quadratic subelement basis functions, thus eliminating the need for fuel-coolant homogenization. Comparisons with reference seven-group Monte Carlo solutions indicate that in the absence of pin-cell homogenization, high-order angular approximations are required to obtain accurate eigenvalues, while the results are substantially less sensitive to the refinement of the finite subelement grids.

Higher order dual Lagrange multiplier spaces�for mortar finite element discretizations

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We construct locally supported and continuous dual basis functions for quadratic finite elements starting from the discontinuous quadratic dual basis functions for the Lagrange multiplier space. In particular, we compare different dual Lagrange multiplier spaces and piecewise linear and quadratic finite elements. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach.

A fixed-grid finite element based enthalpy formulation for generalized phase change problems: role of superficial mushy region

The enthalpy method is primarily developed for studying phase change in a multicomponent material, characterized by a continuous liquid volume fraction ( φ l) vs temperature ( T) relationship. Using the Galerkin finite element method we obtain solutions to the enthalpy formulation for phase change in 1D slabs of pure material, by assuming a superficial phase change region (linear φ l vs T) around the discontinuity at the melting point. Errors between the computed and analytical solutions are evaluated for the fluxes at, and positions of, the freezing front, for different widths of the superficial phase change region and spatial discretizations with linear and quadratic basis functions. For Stefan number ( St) varying between 0.1 and 10 the method is relatively insensitive to spatial discretization and widths of the superficial phase change region. Greater sensitivity is observed at St=0.01, where the variation in the enthalpy is large. In general the width of the superficial phase change region should span at least 2–3 Gauss quadrature points for the enthalpy to be computed accurately. The method is applied to study conventional melting of slabs of frozen brine and ice. Regardless of the forms for the φ l vs T relationships, the thawing times were found to scale as the square of the slab thickness. The ability of the method to efficiently capture multiple thawing fronts which may originate at any spatial location within the sample, is illustrated with the microwave thawing of slabs and 2D cylinders.

Finite element analysis of contaminant transport in groundwater

In this paper, we focus on the development of a finite element model for predicting the contaminant concentration governed by the advective–dispersive equation. In this study, we take into account the first-order degradation of the contaminant to realistically model the transport phenomenon in groundwater. To solve the resulting unsteady advection–diffusion equation with production, a finite element model is constructed, which employs a quadratic basis function to approximate the contaminant concentration. The development of a weighted residuals finite element model involves constructing a biased test function to retain the scheme stability for wide ranges of values of the physical coefficients. In the process of constructing the Petrov–Galerkin finite element model for stability reasons, it is desirable to obtain an acceptable degree of accuracy. The method used to retain stability without loss of accuracy is to approximate the differential equation within the semi-discretization framework. After discretizing the time derivative term using the Euler time-stepping scheme, the resulting ordinary differential equation, which involves only the spatial derivative terms, is solved using the nodally exact finite element model. For better control of the user's specified time step and mesh size, full analysis of the discretization scheme is conducted. In this study, both modified equation analysis and Fourier stability analysis are employed to better understand the proposed semi-discretized Petrov–Galerkin finite element model. Validation of the newly proposed model is accomplished through analysis of the results obtained for several test problems. Some of them are amenable to analytic solutions. The rate of convergence of the employed finite element model then can be obtained.

Numerical analysis of printed superquadric wire loop antennas

The printed superquadric loop antennas, whose geometries including square, rectangular, circular elliptical, and rhombic loops can be represented in a convenient parametric form, are analyzed using the moment method based on quadratic parametric B-spline basis functions. A unified computational formulation for the current distribution and radiation pattern is developed. The effects of the geometric parameters of the superquadric loop on the resonance characteristics of printed wire antennas are investigated. Some useful results are presented.

ROBUST PRECONDITIONERS FOR LINEAR ELASTICITY FEM ANALYSES

SUMMARY This paper deals with two forms of preconditioner which can be easily used with a Conjugate Gradient solver to replace a direct solution subroutine in a traditional engineering nite element package; they are tested in such a package (FINAL) over a range of 2-D and 3-D elasticity problems from geotechnical engineering. Quadratic basis functions are used. A number of modications to the basic Incomplete Choleski [IC(0)] factorization preconditioner are considered. An algorithm to reduce positive o-diagonal entries is shown in numerical experiments to ensure stability, but at the expense of slow convergence. An alternative algorithm of Jennings and Malik is more successful, and a relaxation parameter ! is introduced which can make a further signicant improvement in performance while maintaining stability. A heuristic for determining a near-optimal value of ! is proposed. A second form of preconditioning, symmetrically scaled element by element, due to Bartelt, is also shown to perform robustly over a range of problems; it does not require assembly of the global stiness matrix, and has great potential for parallelization. ? 1997 by John Wiley & Sons, Ltd.

C2-CONTINUOUS ELEMENTS FOR BOUNDARY ELEMENT ANALYSIS

Overhauser's original idea of linearly blending two sets of quadratic C0-continuous basis functions to produce a set of C1-continuous basis functions is employed by linearly blending two sets of quadratic C1-continuous basis functions. The result is a set of eight basis functions which are C2-continuous from element to element and can be used for boundary element analysis where post-processing of the solution is required. Solutions to Laplaces equation in simple geometries are used to demonstrate the accuracy of the solutions obtained with the new elements. These new elements also provide more accurate values for the first and second derivatives of the solution in the tangential direction at all points on a smooth boundary and good approximations at corner points. © 1997 by John Wiley & Sons, Ltd.

The finite volume element method with quadratic basis functions

The paper presents a box scheme with quadratic basis functions for the discretisation of elliptic boundary value problems. The resulting discretisation matrix is non-symmetrical (and also not an M-matrix). The stability analysis is based on an elementwise estimation of the scalar product <A h u h ,u h >. Sufficient conditions placed on the triangles of the triangulation lead to discrete ellipticity. Proof of anO(h 2) error estimate is given for these conditions.

A WEIGHTED LEAST-SQUARES DISCRETE FINITE ELEMENT FORMULATION FOR MODELING THREE-DIMENSIONAL VELOCITY RESPONSE FIELDS MEASURED BY A LASER DOPPLER VIBROMETER

This paper reports initial results on the development of spatial dynamic response models derived from scanning laser Doppler vibrometer (LDV) velocity measurements. A weighted least squares discrete non-parametric finite element formulation is developed to reconstruct the magnitude and phase of the three-dimensional (3-D) complex dynamic response field. A fundamental concept in the formulation is that the steady-state solution for the structural dynamic response contains separable spatial and temporal components of the structure's dynamics. Therefore, the steady-state dynamic response is modeled as a spatially continuous 3-D complex velocity field. The least squares formulation minimises the difference between the measured velocity and the dynamic response field equations. Linear, quadratic, and cubic basis functions are used to define finite elements to model the continuous 3-D dynamic response fields. The ‘effective’ dynamic stiffness (or mechanical impedance) matrices are developed for each element as well as the ‘effective’ nodal forces. To examine the method, analytical test models are defined to simulate the surface velocity response of structures with both in-plane and out-of-plane harmonic vibration. Random and uniformly spaced measurements of the simulated dynamic systems are acquired from four locations relative to the structure. In one case, ‘noise’ is added to the test data set. Application of the weighted least-squares discrete finite element formulation is demonstrated with experimental data acquired by scanning a vibrating composite plate with a LDV. Although the single frequency excitation force on the plate is primarily out-of-plane, evidence of in-plane response is demonstrated by the modal solution.

A new numerical technique for calculating current distributions on curved wire antennas - parametric B-spline finite element method

A new numerical technique is presented for calculating the current distributions on wire antennas with a high degree of curvature. The geometry of the wire antenna is expressed by a parametric equation and the current is directly expanded into a series of quadratic B-spline basis functions of the parameter, so that the truncation error in modeling the curved wire structure is avoided. This algorithm is used to analyze a circular loop antenna, a conical equiangular-spiral antenna, and a four-arm Archimedian spiral antenna to demonstrate its validity and practical value.

Preconditioned conjugate gradient methods for three‐dimensional linear elasticity

AbstractFinite element modelling of three‐dimensional elasticity problems give rise to large sparse matrices. Various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques. Incomplete factorizations based on levels of fill, drop tolerance, and a two‐level hierarchical basis are developed. Various techniques for ensuring that the incomplete factors have positive pivots are presented. Computational tests are carried out for problems generated using unstructured tetrahedral meshes. Quadratic basis functions are used. The performance of the iterative methods is compared to a standard direct sparse matrix solver. Problems with up to 70 000 degrees of freedom and small (≪1) element aspect ratio are considered.

A quadratic Petrov‐Galerkin Solution for kinematic wave overland flow

A Petrov‐Galerkin (PG) finite element method was developed to solve the kinematic wave formulation of the overland flow equations. The resultant model uses quadratic basis functions and test functions that are modified by polynomials of cubic and quartic order, yielding a formulation that includes four PH parameters. The PG model was found to reduce the mean sum of square error of the solution compared to a conventional Bubnov‐Galerkin finite element solution by about a factor of 3 as the Courant number (Cr) approached one. Good results were also achieved with the PG method for problems that resulted in shock formation, which are typical of many applied problems of concern. PG parameters were found to depend strongly upon the Courant number and weakly upon the number of nodes in the system. Polynomial expressions were derived to approximate the PG parameters over the range 0 &lt; Cr &lt; 1. As the number of nodes in the system increased, a single‐parameter version of the model yielded solutions approaching the accuracy of the four‐parameter model.

FIXED LOOP SOURCE EM MODELLING RESULTS USING 2D FINITE ELEMENTS1

AbstractAmong electromagnetic sounding techniques, the Mélos method possesses the specific feature of including an apparent resistivity computation. This acts as a normalizing scheme so that 2D modelling results can be obtained without accounting for a true 3D source. However, in order to get reliable numerical modelling results for a 2D magnetic dipole source, improved algorithms are required in order to apply the standard finite‐element technique: quadratic basis functions must be used in place of linear basis functions, and a more sophisticated method than conventional ones is necessary for properly solving the resulting system of linear equations.Such modelling results have been used to study theoretical responses for the Mélos method in the search for conductive bodies in mineral exploration. Two sets of models are presented and discussed. They show that the typical Mélos response to a conductive target is a bipolar anomaly on the apparent resistivity pseudo‐section, with a conductive pole at low frequency which is centred above the target.

River mass transport linear-quadratic finite element model (RIMTRA)

A linear-quadratic finite element model, RIMTRA for predicting the transport of a single contaminant or BOD-DOD interactions in a polluted river is presented. Streamflow condition is one of steady and uniform or non-uniform flow. Galerkin finite element method with linear and quadratic basis functions; forward differencing for time derivative, and time-weighting factor for implicit control were adopted for solution of the governing differential equations. Model verification and performance evaluation resulted in two test problems: (1) a case of pure advection mass transport in which the effects of varying values of grid spacing and time steps on numerical dispersion were evaluated. Also, predicted results were compared with those of five finite difference codes in order to assess the performance of the proposed model; and (2) BOD-DOD profile computation with predicted results compared to analytical solutions. The results show great promise for the proposed RIMTRA model.

An improved cubic Petrov-Galerkin method for simulation of transient advection-diffusion processes in rectangularly decomposable domains

Abstract A modified cubic Petrov-Galerkin method for solution of multidimensional unsteady advection-diffusion problems is proposed. The method is based on a weighted residual formulation in which the trial space is piecewise linear and the test space is a special piecewise cubic space. The test functions are formed by adding a symmetric cubic perturbation to the piecewise linear basis functions. A frequency-fitting algorithm is used to determine the appropriate magnitude of the cubic perturbation. With this frequency fitted parameter, the method is shown to be superior to standard Galerkin and Petrov-Galerkin methods. Both one- and two-dimensional numerical results are presented. The two-dimensional results are for a rotating flow field and demonstrate that the method performs very well in nonconstant velocity fields. The numerical results demonstrate that the accuracy of the cubic Petrov-Galerkin method is comparable to that of a quadratic basis function Galerkin method, with less than half the computational effort.

Two-Dimensional, Time-dependent Modelling of an Arbitrarily Shaped Ice Mass with the Finite-Element Technique

AbstractThe two-dimensional, time-dependent flow of an arbitrarily shaped ice mass can be successfully modeled with the finite-element technique on a small computer. Methods developed for automatically generating the mesh data greatly simplify the data preparation and optimize the numerical simulations. Using quadratic basis functions permits the flow to be approximated quite adequately by only two element rows (five nodes vertically). Mixed-order basis functions, however, must be used so that numerical oscillations do not set in, and the ends of the ice mass, where the thickness tends to zero, must be treated carefully. Time simulations to a steady-state condition are necessary to test such numerical models adequately.South Cascade Glacier, Washington, is currently close to equilibrium. A bedrock sill dominates the bed topography in the lower half of the glacier, rising to a height of about 20% of the ice thickness. This sill produces a maximum increase in the overall thickness of about 6–7% compared to what the thickness would have been if the sill were not present. Finally, this glacier does not appear to be sliding much, if at all, despite its maritime alpine environment. This could help explain the difficulties encountered when trying to measure sliding and basal water pressures on the same glacier (Hodge, 1979), or it could imply that drag exerted by the valley walls has a significantly greater effect than conventional shape-factor concepts imply.

Two-Dimensional, Time-dependent Modelling of an Arbitrarily Shaped Ice Mass with the Finite-Element Technique

AbstractThe two-dimensional, time-dependent flow of an arbitrarily shaped ice mass can be successfully modeled with the finite-element technique on a small computer. Methods developed for automatically generating the mesh data greatly simplify the data preparation and optimize the numerical simulations. Using quadratic basis functions permits the flow to be approximated quite adequately by only two element rows (five nodes vertically). Mixed-order basis functions, however, must be used so that numerical oscillations do not set in, and the ends of the ice mass, where the thickness tends to zero, must be treated carefully. Time simulations to a steady-state condition are necessary to test such numerical models adequately.South Cascade Glacier, Washington, is currently close to equilibrium. A bedrock sill dominates the bed topography in the lower half of the glacier, rising to a height of about 20% of the ice thickness. This sill produces a maximum increase in the overall thickness of about 6–7% compared to what the thickness would have been if the sill were not present. Finally, this glacier does not appear to be sliding much, if at all, despite its maritime alpine environment. This could help explain the difficulties encountered when trying to measure sliding and basal water pressures on the same glacier (Hodge, 1979), or it could imply that drag exerted by the valley walls has a significantly greater effect than conventional shape-factor concepts imply.

Application of the galefxin finite element method to reactor dynamic problems

Abstract The method of Galerkin is used to obtain a semidiscrete formulation of the time-dependent multigroup diffusion equa-tions which are then integrated in time by the θ-method to obtain a system of algebraic equations. For given initial and boundary conditions, successive estimates of the neutron flux for each energy group and the precursor concentrations for each delayed neutron group are obtained. Since the precursor concentrations are not necessarily continuous across an interface, the global basis functions are treated in a different form than the corresponding ones for the neutron flux. A computer code is implemented using quadratic Lagrange basis functions. Numerical results for a one-dimensional subcritical transient problem show that this method is as accurate as the usual finite difference schemes but considerably more efficient.

Exact Extension to the Infinite Domain for the Vortex-In-Cell Method

The three-dimensional vortex-in-cell method has been extended to allow for the numerical simulation of plane layers of vortical fluid bounded on each side by a semi-infinite domain of unsteady potential flow. Exact boundary conditions for Poisson’s equation for the velocity field were derived for the computational domain enclosing all vorticity-containing fluid. A variational principle with quadratic spline basis functions was used to derive the finite pentadiagonal systems that discretize the differential equations.

Accuracy and convergence of a finite element algorithm for turbulent boundary layer flow

The Galerkin-Weighted Residuals formulation is employed to derive an implicit finite element solution algorithm for the non-linear parabolic partial differential equation system governing turbulent boundary layer flow. Solution accuracy and convergence with discretization refinement are quantized in several error norms using linear and quadratic basis functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equations appears extensible to the non-linear equations characteristic of turbulent boundary layer flow.