scholarly journals Sparsified discrete wave problem involving a radiation condition on a prolate spheroidal surface

2019 ◽  
Author(s):  
Hélène Barucq ◽  
M’Barek Fares ◽  
Carola Kruse ◽  
Sébastien Tordeux
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Classical Method ◽  
Separation Of Variables ◽  
Radiation Condition ◽  
High Order ◽  
Neumann Condition ◽  
Dense Matrix ◽  
Radiation Boundary Condition ◽  
Scattering Problems

Abstract We develop and analyse a high-order outgoing radiation boundary condition for solving three-dimensional scattering problems by elongated obstacles. This Dirichlet-to-Neumann condition is constructed using the classical method of separation of variables that allows one to define the scattered field in a truncated domain. It reads as an infinite series that is truncated for numerical purposes. The radiation condition is implemented in a finite element framework represented by a large dense matrix. Fortunately, the dense matrix can be decomposed into a full block matrix that involves the degrees of freedom on the exterior boundary and a sparse finite element matrix. The inversion of the full block is avoided by using a Sherman–Morrison algorithm that reduces the memory usage drastically. Despite being of high order, this method has only a low memory cost.

Matrix-Free Iterative Methods for Parallel Finite Element Computations in Acoustics

1995 ◽  
Author(s):  
Manish Malhotra ◽  
Peter M. Pinsky
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Large Scale ◽  
Convergence Rates ◽  
Iterative Solution ◽  
Radiation Boundary Condition ◽  
Scattering Problems ◽  
Element Discretization ◽  
Non Local ◽  
Matrix Problems

Abstract In this paper we consider the iterative solution of non-Hermitian, indefinite and complex-valued matrix problems arising from finite element discretization of time-harmonic acoustics problems in exterior domains. The computational model is based on using Galerkin least squares finite element methods for the acoustic fluid, combined with the non-local Dirichlet-to-Neumann map as the radiation boundary condition. The emphasis here is to develop efficient computational procedures that are suitable for parallel iterative solution of large-scale problems. In this context, we develop a low-storage implementation of the non-local DtN map which allows the use of this exact boundary condition without any storage penalties related to its non-local nature. In order to accelerate iterative convergence, we consider a multilevel preconditioning approach based on the h-version of the hierarchical finite element method. Finite element formulations that employ hierarchical shape functions yield better conditioned matrices than formulations based on the usual Lagrange functions. This improved conditioning translates into a faster rate of convergence if projections between nodal and hierarchical basis functions are used to construct the preconditioning operator. We present numerical results for the solution of two-dimensional scattering problems to examine convergence rates that are realized on practical discretizations.

Efficacy of Drilling Degrees of Freedom in the Finite Element Modeling of P-and SV-Wave Scattering Problems

2000 ◽  
Vol 16 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Jaehwan Kim ◽  
Vasundara V. Varadan ◽  
Vijay K. Varadan
Keyword(s):  
Finite Element ◽  
Wave Scattering ◽  
Degrees Of Freedom ◽  
Transverse Component ◽  
P Wave ◽  
Infinite Domain ◽  
Scattering Problems ◽  
Hybrid Finite Element Method ◽  
Infinite Domains ◽  
Drilling Degrees Of Freedom

ABSTRACTThis paper deals with a hybrid finite element method for wave scattering problems in infinite domains. Scattering of waves involving complex geometries, in conjunction with infinite domains is modeled by introducing a mathematical boundary within which a finite element representation is employed. On the mathematical boundary, the finite element representation is matched with a known analytical solution in the infinite domain in terms of fields and their derivatives. The derivative continuity is implemented by using a slope constraint. Drilling degrees of freedom at each node of the finite element model are introduced to make the numerical model more sensitive to the transverse component of the elastodynamic field. To verify the effects of drilling degrees freedom and slope constraints individually, reflection of normally incident P and SV waves on a traction free half space is considered. For P-wave incidence, the results indicate that the use of a slope constraint is more effective because it suppresses artificial reflection at the mathematical boundary. For the SV-wave case, the use of drilling degrees of freedom is effective in reducing numerical error at the irregular frequencies.

A Comparison of Semi-Lagrangian and Lagrange-Galerkin hp-FEM Methods in Convection-Diffusion Problems

2011 ◽  
Vol 9 (4) ◽  
pp. 1020-1039 ◽  
Author(s):  
Pedro Galán del Sastre ◽  
Rodolfo Bermejo
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Second Order ◽  
High Order ◽  
Numerical Results ◽  
Convection Diffusion ◽  
Cpu Time ◽  
Better Than ◽  
Diffusion Problems

AbstractWe perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method. The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom, however, for the same level of accuracy both methods are about the same in terms of CPU time. For polynomials of degree larger than 2, Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU time; specially, for polynomials of degree 8 or larger. Also, we have performed tests on the parallelization of these schemes and the speedup obtained is quasi-optimal even with more than 100 processors.

scholarly journals Isogeometric Implementation of High-Order Microplane Model for the Simulation of High-Order Elasticity, Softening, and Localization

2016 ◽  
Vol 84 (1) ◽  
Author(s):  
Erol Lale ◽  
Xinwei Zhou ◽  
Gianluca Cusatis
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Control Point ◽  
Three Dimensional ◽  
Local Strain ◽  
High Order ◽  
Particle Model ◽  
Displacement Fields ◽  
Displacement Based ◽  
The Continuum

In this paper, a recently developed higher-order microplane (HOM) model for softening and localization is implemented within a isogeometric finite-element framework. The HOM model was derived directly from a three-dimensional discrete particle model, and it was shown to be associated with a high-order continuum characterized by independent rotation and displacement fields. Furthermore, the HOM model possesses two characteristic lengths: the first associated with the spacing of flaws in the material internal structure and related to the gradient character of the continuum; the second associated with the size of these flaws and related to the micropolar character of the continuum. The displacement-based finite element implementation of this type of continua requires C1 continuity both within the elements and at the element boundaries. This motivated the implementation of the concept of isogeometric analysis which ensures a higher degree of smoothness and continuity. Nonuniform rational B-splines (NURBS) based isogeometric elements are implemented in a 3D setting, with both displacement and rotational degrees-of-freedom at each control point. The performed numerical analyses demonstrate the effectiveness of the proposed HOM model implementation to ensure optimal convergence in both elastic and softening regime. Furthermore, the proposed approach allows the natural formulation of a localization limiter able to prevent strain localization and spurious mesh sensitivity known to be pathological issues for typical local strain-softening constitutive equations.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

scholarly journals A Finite Element Approach to the Analysis of Rotating Bladed-Disk Assemblies Coupled With Flexible Shaft

1994 ◽  
Author(s):  
Wen Zhang ◽  
Wenliang Wang ◽  
Hao Wang ◽  
Jiong Tang
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Equation Of Motion ◽  
Coupled Systems ◽  
The Finite Element Method ◽  
Bladed Disk ◽  
Modal Synthesis ◽  
Element Approach ◽  
Final Equation

A method for dynamic analysis of flexible bladed-disk/shaft coupled systems is presented in this paper. Being independant substructures first, the rigid-disk/shaft and each of the bladed-disk assemblies are analyzed separately in a centrifugal force field by means of the finite element method. Then through a modal synthesis approach the equation of motion for the integral system is derived. In the vibration analysis of the rotating bladed-disk substructure, the geometrically nonlinear deformation is taken into account and the rotationally periodic symmetry is utilized to condense the degrees of freedom into one sector. The final equation of motion for the coupled system involves the degrees of freedom of the shaft and those of only one sector of each of the bladed-disks, thereby reducing the computer storage. Some computational and experimental results are given.

scholarly journals A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

2020 ◽  
Vol 20 (4) ◽  
pp. 799-813
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Asymptotic Relation ◽  
Probabilistic Approach ◽  
Random Variable ◽  
High Order ◽  
Relative Accuracy ◽  
The Difference ◽  
New Perspective

AbstractThe aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble–Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements {P_{k}} and {P_{m}} ({k<m}). Then we analyze the asymptotic relation between these two probabilistic laws when the difference {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

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