High-order Finite Element Schemes Research Articles

Time-domain, shallow-water hydroelastic analysis of VLFS elastically connected to the seabed

In order to ensure the safe operation of a VLFS, a combination of mooring, breakwater and other motion reducing systems is employed. In the present work, the transient hydroelastic response of a floating, thin elastic plate, elastically connected to the seabed, is examined. The plate is modelled as an Euler-Bernoulli strip, while the linearized shallow water equations are used for the hydrodynamic modelling. Elastic connectors are approximated by a series of simple spring-dashpot systems positioned along the strip. A higher order finite element scheme is employed for the calculation of the hydroelastic response of the strip-connector configuration over the shallow bathymetry. After the definition of the initial-boundary value problem, its variational form is derived and discussed. Next, on the basis of the aforementioned formulation, an energy balance expression is obtained. The effect of variable bathymetry on the response of a two connector-strip system is examined by means of three seabed profiles featuring a flat bottom, an upslope and a downslope environment. For the flat bottom case, the strip response mitigating effect exerted by the employment of two and three elastic connectors is considered. Finally, by means of the derived energy balance equation, the energy exchange is monitored, providing a valuable insight into the transient phenomena that take place in the studied configurations.

A high order finite element scheme for pricing options under regime switching jump diffusion processes

This paper considers the numerical pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. In an incomplete market structure and using the no-arbitrage pricing principle, the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro-differential equations describing different states of a Markov chain. We develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration. Various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model.

FEM Hydroelastic Models with Application to the Nonlinear Response of Large Floating Bodies in Shallow Wave Conditions

Abstract A higher order finite element scheme is presented for the study of the transient hydroelastic response of a floating, thin, nonlinear strip in shallow wave conditions. First, nonlinear effects are introduced only in the elasticity model, where large deflections and non-negligible normal stress variation in the lateral direction are assumed. The nonlinear beam is initially coupled with the linearized and subsequently with the full nonlinear Shallow Water equations, introducing nonlinearity in both the hydrodynamics and the elasticity model. The effects of the incorporated nonlinear effects are assessed through a numerical example featuring an elevation pulse of increasing steepness.

Analytical and higher order finite element hybrid approach for an efficient simulation of ultrasonic guided waves I: 2D-analysis

In recent years the interest in online monitoring of lightweight structures with ultrasonic guided waves is steadily growing. Especially the aircraft industry is a driving force in the development of structural health monitoring (SHM) systems. In order to optimally design SHM systems powerful and efficient numerical simulation tools to predict the behaviour of ultrasonic elastic waves in thin-walled structures are required. It has been shown that in real industrial applications, such as airplane wings or fuselages, conventional linear and quadratic pure displacement finite elements commonly used to model ultrasonic elastic waves quickly reach their limits. The required mesh density, to obtain good quality solutions, results in enormous computational costs when solving the wave propagation problem in the time domain. To resolve this problem different possibilities are available. Analytical methods and higher order finite element method approaches (HO-FEM), like p-FEM, spectral elements, spectral analysis and isogeometric analysis, are among them. Although analytical approaches offer fast and accurate results, they are limited to rather simple geometries. On the other hand, the application of higher order finite element schemes is a computationally demanding task. The drawbacks of both methods can be circumvented if regions of complex geometry are modelled using a HO-FEM approach while the response of the remaining structure is computed utilizing an analytical approach. The objective of the paper is to present an efficient method to couple different HO-FEM schemes with an analytical description of an undisturbed region. Using this hybrid formulation the numerical effort can be drastically reduced. The functionality of the proposed scheme is demonstrated by studying the propagation of ultrasonic guided waves in plates, excited by a piezoelectric patch actuator. The actuator is modelled utilizing higher order coupled field finite elements, whereas the homogenous, isotropic plate is described analytically. The results of this "semi-analytical" approach highlight the opportunities to reduce the numerical effort if closed-form solutions are partially available.

Dispersive behaviour of high order finite element schemes for the one-way wave equation

We study the ability of high order numerical methods to propagate discrete waves at the same speed as the physical waves in the case of the one-way wave equation. A detailed analysis of the finite element method is presented including an explicit form for the discrete dispersion relation and a complete characterisation of the numerical Bloch waves admitted by the scheme. A comparison is made with the spectral element method and the discontinuous Galerkin method with centred fluxes. It is shown that all schemes admit a spurious mode. The spectral element method is always inferior to the finite element and discontinuous Galerkin schemes; a somewhat surprising result in view of the fact that, in the case of the second order wave equation, the spectral element method propagates waves with an accuracy superior to that of the finite element scheme. The comparative behaviour of the finite element and discontinuous Galerkin scheme is also somewhat surprising: the accuracy of the finite element method is superior to that of the discontinuous Galerkin method in the case of elements of odd order by two orders of accuracy, but worse, again by two orders of accuracy, in the case of elements of even order.

Comparison of different higher order finite element schemes for the simulation of Lamb waves

Structural Health Monitoring (SHM) applications call for both efficient and powerful numerical tools to predict the behavior of ultrasonic guided waves. When considering waves in thin-walled structures, so called Lamb waves, conventional linear or quadratic pure displacement finite elements soon reach their limits. The spatial as well as temporal discretisation, required to obtain good quality results has to be very fine. This results in enormous computational costs (computational time and memory storage requirements) when ultrasonic wave propagation problems are solved in the time domain. To resolve this issue several higher order finite element methods with polynomial degrees p>2 are proposed. The objective of the current article is to develop such higher order schemes and to verify their capabilities with respect to accuracy and numerical performance. To the best of the authors’ knowledge such comparison has not been reported in literature, yet. Specifically, spectral elements based on Lagarange polynomials (SEM), p-elements using the normalized integrals of the Legendre polynomials (p-FEM) and isogeometric elements utilizing non-uniform rational B-splines (NURBS, N-FEM) are discussed in this paper. By solving a two-dimensional benchmark problem, their advantages and drawbacks with respect to Lamb wave propagation are highlighted. The results of the convergence studies are then used to derive guidelines for estimating the optimal element size for a given finite element type and polynomial degree template. These findings serve the purpose to determine the optimal mesh configuration a priori and thus, save a considerable amount of computational effort. The proposed guideline is then tested on a three-dimensional structure with a conical hole showing an excellent agreement with the predicted behaviour.

Follower loads for high order finite elements

AbstractFinite element modelling of hydrostatic compaction where the applied pressure acts normal to the deformed surface requires a geometric nonlinear formulation and follower load terms [1, 5, 7]. These concepts are applied to high order [6] (p‐FEM) elements with hierarchic shape functions. Applying the blending function method allows to precisely describe curved boundaries on coarse meshes. High order elements exhibit good performance even for high aspect ratios and strong distortion and therefore allow an efficient discretization of thin‐walled structures. Since high order finite elements are less prone to locking effects a pure displacement‐based formulation can be chosen.After introducing the basic concept of the p‐version the application of follower loads to geometrically nonlinear high order elements is presented. For the numerical solution the displacement based formulation is linearized yielding the basis for a Newton‐Raphson iteration. The accuracy and performance of the high order finite element scheme is demonstrated by a numerical example. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number

The dispersive properties of high order finite element schemes are analyzed in the setting of the Helmholtz equation, and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order. It is shown that the numerical dispersion displays three different types of behavior, depending on the size of the order of the method relative to the mesh-size and the wave number. Quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes. All estimates are fully explicit and are shown to be sharp. Limits are obtained, where transitions between the different regimes occur, and used to provide guidelines for the selection of the mesh-size and the polynomial order in terms of the wave number so that the dispersion error is controlled.

542 流れ問題に対する高次有限要素法の性能評価

542 流れ問題に対する高次有限要素法の性能評価

Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation

The higher-order finite-element scheme with mass lumping for triangles and tetrahedra is an efficient method for solving the wave equation. A number of lower-order elements have already been found. Here the search for elements of higher order is continued. Elements are constructed in a systematic manner. The nodes are chosen in a symmetric way. Integration rules must be exact up to a certain degree to maintain an overall accuracy that is the same as without mass lumping. First, for given integration degrees, consistent rule structures are derived for which integration formulas are likely to exist. Then, as each rule structure corresponds to a potential element of certain order, the position of element nodes and the integration weights can be found by solving the related system of nonlinear equations. With this systematic approach, a number of new sixth-order triangular elements and a new fourth-order tetrahedral element have been found.

A higher-order finite element scheme for incompressible lubrication calculations

A higher-order finite element scheme is formulated for incompressible lubrication calculations, based on the energy functional of the lubricating system (derived from the variational technique) and the hierarchical approximation concept. The current formulation ensures the pressure continuity across inter-element mating boundaries. Since this is a hierarchical formulation, it facilitates convergence studies of results. Numerical examples are provided to demonstrate the accuracy of the proposed method, the simplicity of modeling, applications, and the convergence characteristics of numerical solutions.

Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations

In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.

The Finite-Element Method for Finding Modes of Dielectric-Loaded Cavities (Short Papers)

A three-dimensional high-order finite-element scheme is proposed for finding the modes of dielectric-loaded cavities of arbitrary shape. New light is shed on the presence of spurious solutions. Results for an empty rectangular box are compared with exact solutions, A loaded rectangular box is also analyzed.

Application of finite element methods to solving immiscible displacement problems with sharp fronts (abstract)

Many reservoir engineering problems involve solving fluid flow equations, whose solutions are characterized by sharp fronts and low dispersion levels. This is particularly important in tracking small slugs which are characteristic of chemical floods, polymer floods, first and multiple contact hydrocarbon miscible displacements and most thermal processes. The use of finite difference approximations to solve these problems when low dispersion levels and small slugs need to be accurately modeled is prohibitively expensive. In two dimensions, as many as 500 grid blocks on a side would be required in order to obtain the necessary accuracy. However, the use of high order finite element schemes are shown to be a very effective means of economically solving this problem. This paper presents some numerical results of a research project recently completed which has demonstrated that high order finite element methods can be used to solve difficult reservoir engineering problems where finite difference approximations would be prohibitively expensive.