Finite Element-Based Algorithms To Make Cuts for Magnetic Scalar Potentials: Topological Constraints and Computational Complexity - Abstract

We develop finite element data structures for T‐splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so‐called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T‐junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T‐splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.

: The principal goals of the project are; The derivation of a uniformly stable boundary element formulations in the equations valid for the whole range of wave numbers k. The formulation should be compatible with the the C(O) regularity assumption for finite element approximations of the structure. The development of C(O) finite element approximations for one-, two-, and three- dimensional geometries. This part of the project included such fundamental issues as the development of new hp finite element data structures and constrained approximations for coupled boundary element/finite element problems. The development of a coupled finite element/boundary element code and the verification of expected rates of convergence. The derivation, implementation, and verification of an a posteriori error estimate for boundary element methods. The development of adaptive strategies. Having obtained a local error estimate, these strategies would provide for reducing mesh size or increasing spectral order so as to achieve a target error with a number of unknowns. The solution of selected problems. These are designed to demonstrate the effectiveness of the methodology on classical benchmark problems and, ultimately, on realistic scattering problems of interest in Naval research. The verification of the applicability of the hp approximation to structural mechanics, including bulky structures such as bulkheads, but also possibly shells, thin beams and plates. The development of a mathematical foundation for transient acoustics, including studying mathematical properties of linear hyperbolic systems associated with transient acoustics, the development of new transient hp-adaptive schemes, the development of high order transient solvers and associated a posteriori error estimates, and associated adaptive strategies.

We present an algorithm for computing the closest point, transform to an explicitly described manifold on a rectilinear grid in low dimensional spaces. The closest point transform finds the closest point on a manifold and the Euclidean distance to a manifold for the points in a grid. We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm solves the eikonal equation |∇u| = 1 with the method of characteristics. For many problems, the computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold. Many query problems can be aided by using orthogonal range queries (ORQ). There are several standard data structures for performing ORQ's in 3-D, including kd-trees, octrees, and cell arrays. We develop additional data structures based on cell arrays. We study the characteristics of each data structure and compare their performance. We present a new algorithm for solving the single-source, non-negative weight, shortest-paths problem. Dijkstra's algorithm solves this problem with computational complexity O ((E + V)log V) where E is the number of edges and V is the number of vertices. The new algorithm, called Marching with a Correctness Criterion (MCC), has computational complexity O (E + RV), where R is the ratio of the largest to smallest edge weight. Sethian's Fast Marching Method (FMM) may be used to solve static Hamilton-Jacobi equations. It has computational complexity O (N log N), where N is the number of grid points. The FMM has been regarded as an optimal algorithm because it is closely related to Dijkstra's algorithm. The new shortest-paths algorithm discussed above can be used to develop an ordered, upwind, finite difference algorithm for solving static Hamilton-Jacobi equations. This algorithm requires difference schemes that difference not only in coordinate directions, but in diagonal directions as well. It has computational complexity O(RN), where R is the ratio of the largest to smallest propagation speed and N is the number of grid points.

For electrical impedance tomography (EIT) of brain, the use of anatomically accurate and patient-specific finite element (FE) mesh has been shown to confer significant improvements in the quality of image reconstruction. But, given the lack of a rapid method to achieve the accurate anatomic geometry of the head, the generation of patient-specifc mesh is time-comsuming. In this paper, a m odified fuzzy c-means algorithm based on no n-local means method is performed to implement the segmentation of different layers in the head based on head CT images. This algorithm showed a better effect, especially an accurate recognition of the ventricles and a suitable performance dealing with noise. And the FE mesh established according to the segmentation results is valid ated in computational simulation. So a rapid practicable method can be provided for the generation of patient-specific FE mesh of the human head that is suitable for brain EIT. Keyword: Computerized tomography, image segmentation, mesh generation, electrical impedance tomography

Hybrid finite element methods (HFEM) as used in the present paper refer to a coupling of the finite element method (FEM) applied to Maxwell's equations in differential form and the method of moments (MOM) applied to Maxwell's equations in integral form. This coupling allows the strengths of each method to supplant the other. The finite element method models penetrable materials and complicated geometry, while the method of moments provides the exact radiation boundary condition where the finite element mesh is terminated. In the paper a three-dimensional HFEM implementation is presented. As in Boyse and Seidl (1991), the finite element formulation is coupled to a highly efficient BOR MOM code which provides the exact near field radiation condition. The scalar and vector potential formulation completely eliminates spurious modes when using nodal based elements, allowing iterative methods to rapidly converge and the computation of very accurate internal fields. However, calculations involving models with sharp corners initially produced very inaccurate results. A new element (Boyse et al., 1992 and Bardi et al., 1993) was used along corners and edges which combines both nodal and edge basis functions. The addition of these new elements produces very accurate results while retaining conventional nodal elements throughout most of the model.

Sensitivity analysis with the modified Heaviside function for the optimal layout design of multi-component systems

Toward a universal [formula omitted] adaptive finite element strategy, part 1. Constrained approximation and data structure

This paper introduces four graph orientation problems named Maximize W -Light, Minimize W -Light, Maximize W -Heavy, and Minimize W -Heavy, where W can be any fixed non-negative integer. In each of these problems, the input is an undirected graph G and the objective is to assign a direction to each edge in G so that the number of vertices with outdegree at most W or at least W in the resulting directed graph is maximized or minimized. We derive a number of results on the computational complexity and polynomial-time approximability of these problems for different values of W and various special classes of graphs. In particular, we show that Maximize 0-Light and Minimize 1-Heavy are equivalent to Maximum Independent Set and Minimum Vertex Cover, respectively, so by allowing the value of W to vary, we obtain a new, natural generalization of the two latter problems.KeywordsPlanar GraphUndirected GraphVertex CoverMaximum CliqueChordal GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The optimal detection for coded system requires the use of a maximum a posteriori (MAP) detection. A list sphere detector (LSD) can be used to approximate the MAP detector. Depending on the used list size, LSD provides a tradeoff between the performance and the computational complexity. The LSD output candidate list is used to calculate the approximation of the probability log-likelihood ratio (LLR) of each transmitted bit. The list should be large enough and it should include at least one candidate for both possible bits for good approximation. The use of a small list size causes inaccurate and, especially, very large LLRs that prevent the decoder from correcting the falsely detected signals and, thus, degrades performance. We study the effect of the LLR clipping to the performance and complexity of the LSD algorithm. We show that by limiting the dynamic range of the LLR the required LSD list size can be decreased, and, thus, the complexity of the algorithms is decreased. The optimal dynamic range values for LLR clipping are determined and the effect of the clipping to the complexity of the LSD algorithms is analyzed.

This paper considers the issues of the reduction in the computational complexity of the numerical problem and the requirements of the computer memory size when calculating the power of the eddy currents in a hydrogen fuel cell based on a proton exchange membrane. The study was performed on a model problem based on a geometric pattern of a hydrogen fuel cell of typical dimensions and characteristics operating at a nominal load in steady-state conditions. The power of the eddy currents was calculated using a numerical model based on a tetrahedral finite element mesh and a mathematical model of a quasi-stationary electromagnetic field in the classical formulation through the vector magnetic potential. The requirements for the minimum degree of the geometric model approximation by a finite element mesh to achieve mesh stability of the computational problem results were defined. The paper considers issues of finite element order selection, the method of ordering, and the solution of the systems of linear algebraic equations (SLAEs). It shows the techniques for determining the accuracy of the calculated SLAE solution obtained by the direct method, as well as the effectiveness of the low-rank approximation of the SLAE to reduce the computational complexity of the computational problem solution and reduce the requirements for the amount of computer memory needed, considering the reduced accuracy of the SLAE solutions.

SummaryIn this work, a reduced‐order model based on adaptive finite element meshes and a correction term obtained by using an artificial neural network (FAN‐ROM) is presented. The idea is to run a high‐fidelity simulation by using an adaptively refined finite element mesh and compare the results obtained with those of a coarse mesh finite element model. From this comparison, a correction forcing term can be computed for each training configuration. A model for the correction term is built by using an artificial neural network, and the final reduced‐order model is obtained by putting together the coarse mesh finite element model, plus the artificial neural network model for the correction forcing term.The methodology is applied to nonlinear solid mechanics problems, transient quasi‐incompressible flows, and a fluid‐structure interaction problem. The results of the numerical examples show that the FAN‐ROM is capable of improving the simulation results obtained in coarse finite element meshes at a reduced computational cost.

The paper deals with the application of an adaptive, hierarchic‐iterative finite element technique to solve two‐dimensional electromechanical boundary value problems with impermeable cracks in piezoelectric plates. In order to compute the dielectric and mechanical intensity factors, the interaction integral technique is used. The iterative finite element solver takes advantage of a sequence of solutions on hierarchic discretizations. Based on an a posteriori error estimation, the finite element mesh is locally refined or coarsened in each step. Two crack configurations are investigated in an infinite piezoelectric plate: A finite straight crack and a finite kinked crack. Fast convergence of the numerical intensity factors to the corresponding analytical solution is exemplarily proved during successive adaptive steps for the first configuration. Similar tendency can be observed for the second configuration. Furthermore, the computed intensity factors for the kinks are found to coincide well with the corresponding analytical values. In order to simulate the kinks spreading from a straight crack, the finite element mesh is modified automatically with a specially developed algorithm. This forms the basis for a fully adaptive simulation of crack propagation. Copyright © 2009 John Wiley & Sons, Ltd.

High precision magnetic field computer programs are required to accurately calculate the multipole content and magnetic field strength associated with a given accelerator magnet design. In this work, a specialized version of the finite element computer program WEMAP (Westinghouse Electric and Magnetic Analysis Program) is applied to field harmonic calculations. The computer program relies on a higher-order finite element formulation and high-precision computer representation to minimize calculation error. Several calculations are made for a SSC CDM cross section design. The effects of mesh fineness and finite element order are investigated by examining finite element grids with increasing mesh density and element order. A key feature of this computer program is the capability to minimize the truncation error associated with a finite element mesh by increasing the order of the finite element interpolating polynomial. Finite elements of order one through six are incorporated. Inaccuracies which may occur using the lower-order finite elements with a coarse mesh are identified in this paper. Care must be taken in meshing the yoke iron region to accurately account for field harmonics due to nonuniform iron permeability. Methods of extracting the spatial harmonics from the finite element solution are discussed. The field strength and harmonics are calculated for a full-section model of a CDM magnet cross section.

Three-dimensional cellular automaton-finite element modeling of solidification grain structures for arc-welding processes

An efficient contact algorithm for the interaction of material particles with finite elements