Optimal balance between mass and smoothed stiffness in simulation of acoustic problems

The Finite Element Method (FEM) is known to behave overly-stiff, which leads to an imbalance between the mass and stiffness matrices within discretized systems. In this work, for the first time, a model is developed that provides optimal balance between discretized mass and smoothed stiffness—the mass-redistributed alpha finite element method (MR-αFEM). This new method improves on the computational efficiency of the FEM and Smoothed Finite Element Methods (S-FEM). The rigorous research conducted ensures that stiffness with the parameter, α, optimally matches the mass with a flexible integration point, q. The optimal balance system significantly reduces the dispersion error of acoustic problems, including those of single and multi-fluids in both time and frequency domains. The excellent properties of the proposed MR-αFEM are validated using theoretical analyses and numerical examples.     

FEM-BEM coupling with non-conforming interfaces

In order to treat wave propagation phenomena in coupled domains, a combined approach of Finite Element Methods (FEM) and Boundary Element Methods (BEM) is presented. The coupling is done within the framework of Tearing and Interconnecting methods (FETI/BETI), which are special non-overlapping domain decomposition methods. The coupling conditions are incorporated in a weak sense, which allows non-conforming interface discretization, i.e., the Mortar Method is used. A numerical example is given to verify the algorithm. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A discontinuous Galerkin Finite Element Method for linear elasticity using a mixed integration scheme to circumvent shear-locking

Shear-locking is a significant problem when using the Finite Element Method (FEM) with standard lower order Langrangian shape functions. To avoid this effect we introduce the discontinuous Galerkin Finite Element Method (dG-FEM) for linear elasticity. In addition to a standard integration scheme for the dG-FEM we develop a mixed integration scheme and compare both with established finite elements using a bending dominated benchmark. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sobre la utilización de códigos de elementos finitos basados en mallados cartesianos en optimización estructural

The structural shape optimization is an iterative process built up by a higher level, which proposes the geometries to analyze, and a lower level which is in charge of analyzing, numerically, their structural response, usually by means of the Finite Element Method (FEM). These techniques normally report notorious advantages in an industrial environment, but their high computational cost is the main drawback. The efficiency of the global process requires the efficiency of both levels. This work focuses on the improvement of the efficiency of the lower level by using a methodology that uses a 2D linear elasticity code based on geometry-independent Cartesian grids, combined with FEM solution and recovery techniques, adapted to this framework. This mesh type simplifies the mesh generation and, in combination with a hierarchical data structure, reuses a great calculus amount. The recovery technique plays a double role: a) it is used in the Zienkiewicz-Zhu type error estimators allowing to quantify the FEM solution quality to guide the h-adaptive refinement process which minimizes the computational cost for a given accuracy; and b) it provides a solution, more accurate than the FEM one, that can be used. The numerical results, which include a comparative with a commercial code, show the effect of the proposed methodology improving the efficiency in the optimization process and in the solution quality.     

A Discrete Element Approach for Wave Propagation in Thin Rods

Wave propagation and evoked side effects, such as material failure, are important parts of analysis of dynamically exposed structures like buildings or machinery. One of the numerical analysis tools for wave propagation is the well-known Finite Element Method (FEM) with its impressive performance but also with the drawback of not being able to model easily material failure, discontinuities and contacts. In contrast, the Discrete Element Method (DEM) is capable to describe these effects on a meso-scale more easily. The here selected concept is based on a DEM particle which is considered as deformable, and can establish and remove lasting bonds with other particles. The simulation example used is a thin rod that has been an extensive numerical and experimental research subject since the last century. A longitudinal wave is excited within the thin rod by simulating an impact on one end of the rod in a free-free configuration. It is found, that the simulation data, the velocity profile and the resulting displacement at the end of the rod, are in good agreement with experimental obtained data. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison of NRBCs for PUFEM in 2D Helmholtz problems at high wave numbers

In this work, exact and approximate non-reflecting boundary conditions (NRBCs) are implemented with the Partition of Unity Finite Element Method (PUFEM) to solve short wave scattering problems governed by the Helmholtz equation in two dimensions. By short wave problems, we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer. Various NRBCs are implemented and a comparison of their performance is carried out based on the accuracy of the results, ease of implementation and computational cost. The aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number.     

Convergence analysis of a finite volume method via a new nonconforming finite element method

The article is devoted to the study of convergence properties of a Finite Volume Method (FVM) using Voronoi boxes for discretization. The approach is based on the construction of a new nonconforming Finite Element Method (FEM), such that the system of linear equations coincides completely with that for the FVM. Thus, by proving convergence properties of the FEM, we obtain similar ones of the FVM. In this article, the investigations are restricted to the Poisson equation. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:213–231, 1998     

On Neumann - Dirchlet coupling strategy of Finite Element and Boundary Element method in elastodynamics

The Finite Element Method (FEM) and the Boundary Element Method (BEM) are the most used numerical tools for solid mechanics analysis. Each one of these methods has advantages and drawbacks in different cases. In order to take advantage of both methods, a nonoverlapping domain decomposition method FEM - BEM in elastodynamics is presented. The domain is divided in two subdomains and each one of them is analyzed separately and only the interface information is exchanged. An iterative Neumann - Dirchlet algorithm with relaxation is used, to get continuity and the equilibrium conditions at the interface. The FEM time integration is carried out using the Newmark's method and the BEM approach in time domain is based in the Convolution Quadrature Method developed by Lubich. Numerical examples are presented to show agreement with other available numerical results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study of Mode Conversion at Defects in Rope Structures using Ultrasonic Waves

Ultrasonic waves travel in rope structures over long distances as guided waves, allowing for effective health monitoring. In order to localize and characterize defects, an exact knowledge of the propagation, reflection, and transmission properties of the ultrasonic waves is required. These properties can be obtained using the Finite Element Method by modeling a segment of the periodic waveguide with a periodicity condition. The solution of the corresponding eigenvalue problem leads to all propagating modes of the waveguide as well as locally generated evanescent modes. The Boundary Element Method (BEM) is used in combination with the Finite Element Method for characterizing the wave propagation. The mode conversion at discontinuities, such as cracks or notches, can be subsequently described by reflection and transmission coefficients. The simulation results are the corresponding coefficients as a function of frequency and enable the selection of adequate modes for an effective defect detection. Additionally, it is demonstrated that along with the localization of cracks, conclusions about the crack geometry can be made with the help of reflection and transmission coefficients. The reliability and numerical accuracy of the simulation results are verfied by comparison with experimental findings. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

尖劈吸波体与导引仿真用微波暗室的性能研究

首先,针对尖劈形状吸波体的性能问题,给出了直接计算法和基于镜像模型的方法,并对其进行了对比计算与仿真.其次,对于微波暗室的性能研究,针对不同的复杂度要求,建立了两种数学模型—射线追踪(Ray Tracing)模型和基于Markov链的有限元(FEM,Finite Element Model)模型.建模过程和仿真结果表明,Ray Tracing模型的计算复杂度较低,但电磁波"镜面反射"的假设过于理想,模型较为粗糙,只能用于粗略模拟实际情况.而基于Markov链的FEM模型较Ray Tracing模型更加精确.同时,相比于传统的具有高计算复杂度的FEM模型,基于Markov链的FEM模型计算更加简便,利于计算机仿真实现,而且不降低FEM模型的精确度,可以精确模拟实际情况.     

Simulation of 2D linear crack growth under constant load using GFVM and two-point displacement extrapolation method

A new approach to model two-dimensional linear crack propagation, based on the Galerkin Finite Volume Method (GFVM), is proposed. The displacement field is calculated using the GFVM method by solving two-dimensional equilibrium equations on an unstructured triangular mesh. An essential feature of this method is that it does not require matrix operations; hence, it obviously reduces computation time. The Two-Point Displacement Extrapolation (TPDE) technique is employed to calculate Stress Intensity Factors (SIFs). The accuracy of the structural solver that has been developed is evaluated using five test cases. In the first example, a Timoshenko cantilever beam, carrying an end point load, is analyzed. In the second and third examples, stress intensity factors are computed for edge and inner crack development in plates under transient loading. The GFVM results are then compared with their counterparts that resulted from the Explicit Finite Element Method (E-FEM). The comparison indicates that the FVM has an accuracy close to E-FEM, whereas the FVM drastically reduces the computational time. A case study is conducted to simulate the gradual propagation of crack. The results computed by the numerical simulation presented are in excellent agreement with the corresponding results from the analytical solution as well as experimental measurements.     

The use of generalized finite difference method in perfectly matched layer analysis

This study deals with the use of Generalized Finite Difference Method (GFDM) in Perfectly Matched Layer (PML) analysis. There are two options for performing PML analysis. First option is to express PML equations in terms of real coordinates of the points in actual (real) PML region; the second is to use governing equations (expressed in terms of complex stretching coordinates) as they are in complex PML region. The first option is implemented in this study; the implementation of the second option is under way and will be reported in another study. For the integration of PML equations, the use of GFDM is proposed. Finally, the suggested procedure is assessed computationally by considering the compliance functions of surface and embedded rigid strip foundations. GFDM with PML results are compared to those obtained by using Finite Element Method (FEM) with PML and Boundary Element Method (BEM). Excellent matches in results showed the reliability of the proposed procedure in PML analysis.     

Computational Modeling of Liquid Droplets Moving on Rough Surfaces

We present a de-coupled approach for computational modeling of liquid droplets moving on rough substrate surfaces. The computational model comprises solving the membrane deformation problem and the fluid flow problem in a segregated manner. The droplet shape is first computed by solving the Young-Laplace equation where contact constraints, due to the droplet-substrate contact, are applied through the penalty method [1]. The resulting configuration constitutes the domain for the fluid flow problem, where the bulk fluid behavior is modeled by the unsteady Stokes' flow model expressed in Arbitrary Lagrangian-Eulerian (ALE) framework. The entire analysis is performed in the framework of Finite Element Method (FEM). Application of the approach to the case of a droplet moving on a rough surface is presented as an example. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic Wave Propagation in Porous Media Semi-Infinite Domains

The problem of dynamic wave propagation in semi-infinite domains is of great importance, especially, in subjects of applied mechanics and geomechanics, such as the issues of earthquake wave propagation in an infinite half-space and soil-structure interaction under seismic loading. In such problems, the elastic waves are supposed to propagate to infinity, which requires a special treatment of the boundaries in initial boundary-value problems (IBVP). Saturated porous materials, e. g. soil, basically represent volumetrically coupled solid-fluid aggregates. Based on the continuum-mechanical principles and the established macroscopic Theory of Porous Media (TPM) [1, 2], the governing balance equations yield a coupled system of partial differential equations (PDE). Restricting the discussion to the isothermal and geometrically linear case, this system comprises the solid and fluid momentum balances and the overall volume balance, and can be conveniently treated numerically following an implicit monolithic approach [3]. Therefore, the equations are firstly discretised in space using the mixed Finite Element Method (FEM) together with quasi-static Infinite Elements (IE) at the boundaries that represent the extension of the domain to infinity [4], and secondly in time using an appropriate implicit time-integration scheme. Additionally, a stable implementation of the Viscous Damping Boundary (VDB) method [5] for the simulation of transient waves at infinity is presented, which implicitly treats the damping boundary terms in a weakly imposed sense. The proposed algorithm is implemented into the FE tool PANDAS and tested on a two-dimensional IBVP. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Higher order Trefftz-like Finite Element Method on meshes with L-shaped elements

In 2009, the BEM-based FEM was introduced as an numerical approach for the treatment of boundary value problems. It is a Finite Element Method (FEM) that uses Trefftz-like basis functions which are defined to fulfil the underlying differential equation locally and which are treated by means of Boundary Element Methods (BEM). Due to the implicit definition of basis functions, this approach is applicable on general polygonal and polyhedral meshes and yields conforming approximations. The elements of the discretization do not necessarily have to be convex. After a review of the recent development of higher order basis functions the method is applied to a model problem on a sequence of meshes with L-shaped elements. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

求积元法在金融工程计算领域应用初探

金融工程领域的大量实际问题最终都可归结为对随机微分方程(组)的求解.针对金融工程计算领域涉及到的静态一维问题,首次将求积元方法应用于非自伴随微分方程的求解.建立了相应的求积元方法计算单元.对典型问题进行计算,并与解析解、有限差分解、有限元解分别进行对比.结果表明,求积元法是一种简单准确高效的数值方法,可进一步用于金融工程计算领域动态问题、二维问题的计算分析.     

Numerical computation of dispersion relations in wave guides

In this paper a numerical approach, based on the Scaled Boundary Finite Element Method (SBFEM), is described to obtain dispersion relations for propagating modes in wave guides. While the formulation is developed for plate structures, it can easily be extended to wave guides with arbitrary cross-section. The cross-section is discretized in the Finite Element sense while all equations remain analytical in the direction of propagation. The wave numbers of all propagating modes are obtained as the solutions of a standard eigenvalue problem. The group velocities can be calculated accurately as the eigenvalue derivatives. The use of higher-order elements drastically increases the efficiency and accuracy of the computation. This approach can be used for wave guides with arbitrary distribution of material parameters. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dispersion in Cylindrical Waveguides with Uncertain Parameters

In order to localize cracks in cylindrical structures using guided waves, precise knowledge of the wave speeds is crucial. Instead of basing calculations on crisp parameters, for this Structural Health Monitoring application, uncertainty in parameters is handled by representing parameters as fuzzy numbers and applying the Transformation Method. For calculating dispersion curves, the Waveguide Finite Element Method is used for each parameter set. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enriching the Finite Element Method with meshfree techniques in structural mechanics

The automatic generation of meshes for the Finite Element method can be an expensive computational burden, especially in structural problems with localized stress peaks. The use of meshless methods can address such an issue, as these techniques do not require the existence of an underlying connection among the nodes selected in a general domain. However, a thoroughly meshfree technique can be computationally quite expensive. Usually, the most expensive tasks rely on identifying the nodal contacts and computing the Galerkin integrals. In this thesis we advance a novel hybrid technique that blends Finite Elements with the Meshless Local Petrov-Galerkin method with the aim at exploiting the most attractive properties of each procedure. The idea relies on the use of the Finite Element Method to compute a background solution that is locally improved by enriching the approximating space with the basis functions associated to a few meshless nodes, thus taking advantage of the flexibility ensured by the use of particles disconnected from an underlying grid. Adding the meshless particles only where needed avoids the cost of mesh refining, or even of re-meshing, without the prohibitive burden of a thoroughly meshfree approach. In particular, two enriching methods are introduced and discussed, with applications in structural mechanics. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semi-analytic actuating and sensing in regular and irregular MEMs,single and assembled micro cantilevers

The static and dynamic behavior of regular and irregular single and assembled micro cantilever probes (ACPs) have been analysed. Various points and distributed loadings are considered. Since the applications of Micro Electro Mechanical Systems (MEMS) are not limited to an especial boundary condition, two semi-analytical approaches, named the generalized differential quadrature and generalized differential quadrature element methods (GDQM and GDQEM) have been used for regular and irregular MEMS, respectively. With less computational cost, it has been clearly demonstrated that these methods are more accurate than common numerical methods such as the Finite Element Method (FEM). For probable cases, proposed approaches have been validated with the exact Green’s function method. Then, considering the various composite lamination configurations (angle/cross), the effects of the electromechanical loading on the nano steering devices have been introduced. At the end, the solution challenges for scanning (sensing) the especial micro and nano profiles has been discussed and as a general case, a nano gear has been studied. The phase plane approach shows the probability of solution for various configurations and suggests the best for more stability.