Showing papers on "Finite element method published in 2017"

Energy and Finite Element Methods in Structural Mechanics : SI Units Edition

Abstract: The first two parts - ''Foundations of Solid Mechanics and Variational Methods'' and ''Structural Mechanics'' - develop a foundation in variational calculus and energy methods before progressing to the third section, which examines the finite element method and its application to stress, plate, torsion, stability, and dynamics problems. Throughout, the book makes finite elements more understandable in terms of fundamentals; provides the background needed to extrapolate the finite element method to areas of study other than solid mechanics; and shows how to derive working equations of structural mechanics through variational principles and to understand the limits of validity of these equations. New to the Second Edition are chapters on matrix methods for trusses, finite element methods for plane stress problems, and finite element methods for plates and elastic stability.

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

Abstract: The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $H(div)$-conforming finite ...

hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

Abstract: An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analysed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (P_p-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a P_p-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.

Finite Elements Theory Fast Solvers And Applications In Solid Mechanics

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An efficient 3D finite element method model based on the T–A formulation for superconducting coated conductors

Abstract: An efficient three dimensional (3D) finite element method numerical model is proposed for superconducting coated conductors. The model is based on the T–A formulation and can be used to tackle 3D computational challenges for superconductors with high aspect ratios. By assuming a sheet approximation for the conductors, the model can speed up the computational process. The model has been validated by established analytical solutions. Two examples with complex geometries, which can hardly be simulated by the 2D model, are given. The model could be used to characterise and design large-scale applications using superconducting coated conductors, such as high field magnets and other electrical devices.

Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates

Abstract: By means of Non-Uniform Rational B-Splines (NURBS) curves, it is possible to describe arbitrary shapes with holes and discontinuities. These peculiar shapes can be taken into account to describe the reference domain of several nanoplates, where a nanoplate refers to a flat structure reinforced with Carbon Nanotubes (CNTs). In the present paper, a micromechanical model based on the agglomeration of these nanoparticles is considered. Indeed, when this kind of reinforcing phase is inserted into a polymeric matrix, CNTs tend to increase their density in some regions. Nevertheless, some nanoparticles can be still scattered within the matrix. The proposed model allows to control the agglomeration by means of two parameters. In this way, several parametric studies are presented to show the influence of this agglomeration on the free vibrations. The considered structures are characterized also by a gradual variation of CNTs along the plate thickness. Thus, the term Functionally Graded Carbon Nanotubes (FG-CNTs) is introduced to specify these plates. Some additional parametric studies are also performed to analyze the effect of a mesh distortion, by considering several geometric and mechanical configurations. The validity of the current methodology is proven through a comparative assessment of our results with those available from the literature or obtained with different numerical approaches, such as the Finite Element Method (FEM). The strong form of the equations governing a plate is solved by means of the Generalized Differential Quadrature (GDQ) method.

Topology optimization of multi-material negative Poissons ratio metamaterials using a reconciled level set method

Abstract: Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poissons Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the MerrimanBenceOsher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poissons ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well. A multi-material topology optimization approach exploiting the reconciled level-set method.The boundary of each individual material is evolved with a single level set function.Multiple level set functions are reconciled with the MerrimanBenceOsher (MBO) operator.Both 2D and 3D multi-material designs were obtained and used for validate the proposed method.

A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

Abstract: This paper deals with the integral version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the Holder regularity of the data. By relying on these results, optimal order of convergence for the standard linear finite element method is proved for quasi-uniform as well as graded meshes. Some numerical examples are given showing results in agreement with the theoretical predictions.

A fully coupled method for massively parallel simulation of hydraulically driven fractures in 3-dimensions

Abstract: Summary This paper describes a fully coupled finite element/finite volume approach for simulating field-scale hydraulically driven fractures in three dimensions, using massively parallel computing platforms. The proposed method is capable of capturing realistic representations of local heterogeneities, layering and natural fracture networks in a reservoir. A detailed description of the numerical implementation is provided, along with numerical studies comparing the model with both analytical solutions and experimental results. The results demonstrate the effectiveness of the proposed method for modeling large-scale problems involving hydraulically driven fractures in three dimensions. © 2016 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.

Finite element quasi-interpolation and best approximation

Abstract: This paper introduces a quasi-interpolation operator for scalar-and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator is stable in L^1 , is a projection, whether homogeneous boundary conditions are imposed or not, and, assuming regularity in the fractional Sobolev spaces W^{s,p} where p ∈ [1, ∞] and s can be arbitrarily close to zero, gives optimal local approximation estimates in any L^p-norm. The theory is illustrated on H^1-, H(curl)-and H(div)-conforming spaces.

Arbitrary order 2D virtual elements for polygonal meshes: part II, inelastic problem

Abstract: The present work deals with the formulation of a virtual element method for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II (Artioli et al. in Comput Mech, 2017) the method is extended to material nonlinearity, considering different inelastic responses of the material. In particular, in part I a standardized procedure for the construction of all the terms required for the implementation of the method in a computer code is explained. The procedure is initially illustrated for the simplest case of quadrilateral virtual elements with linear approximation of displacement variables on the boundary of the element. Then, the case of polygonal elements with quadratic and, even, higher order interpolation is considered. The construction of the method is detailed, deriving the approximation of the consistent term, the required stabilization term and the loading term for all the considered virtual elements. A wide numerical investigation is performed to assess the performances of the developed virtual elements, considering different number of edges describing the elements and different order of approximations of the unknown field. Numerical results are also compared with the one recovered using the classical finite element method.

Unconditionally Convergent $L1$-Galerkin FEMs for Nonlinear Time-Fractional Schrödinger Equations

Abstract: In this paper, a linearized $L1$-Galerkin finite element method is proposed to solve the multidimensional nonlinear time-fractional Schrodinger equation. In terms of a temporal-spatial error splitting argument, we prove that the finite element approximations in the $L^2$-norm and $L^\infty$-norm are bounded without any time-step size conditions. More importantly, by using a discrete fractional Gronwall-type inequality, optimal error estimates of the numerical schemes are obtained unconditionally, while the classical analysis for multidimensional nonlinear fractional problems always required certain time-step restrictions dependent on the spatial mesh size. Numerical examples are given to illustrate our theoretical results.

Parameter-robust discretization and preconditioning of Biot's consolidation model

Abstract: Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters range over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well-behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in nonstandard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.

Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system

Abstract: In this paper, we present a novel second order in time mixed finite element scheme for the Cahn–Hilliard–Navier–Stokes equations with matched densities. The scheme combines a standard second order Crank–Nicolson method for the Navier–Stokes equations and a modification to the Crank–Nicolson method for the Cahn–Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn–Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $$\ell ^\infty \left( 0,T;L^\infty \right) $$ and the discrete chemical potential bounded in $$\ell ^\infty \left( 0,T;L^2\right) $$ , for any time and space step sizes, in two and three dimensions, and for any finite final time T. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.

Theory and computation of higher gradient elasticity theories based on action principles

Abstract: In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packages—by using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect.

Design and analysis of strut-based lattice structures for vibration isolation

Abstract: This paper presents the design, analysis and experimental verification of strut-based lattice structures to enhance the mechanical vibration isolation properties of a machine frame, whilst also conserving its structural integrity. In addition, design parameters that correlate lattices, with fixed volume and similar material, to natural frequency and structural integrity are also presented. To achieve high efficiency of vibration isolation and to conserve the structural integrity, a trade-off needs to be made between the frame’s natural frequency and its compressive strength. The total area moment of inertia and the mass (at fixed volume and with similar material) are proposed design parameters to compare and select the lattice structures; these parameters are computationally efficient and straight-forward to compute, as opposed to the use of finite element modelling to estimate both natural frequency and compressive strength. However, to validate the design parameters, finite element modelling has been used to determine the theoretical static and dynamic mechanical properties of the lattice structures. The lattices have been fabricated by laser powder bed fusion and experimentally tested to compare their static and dynamic properties to the theoretical model. Correlations between the proposed design parameters, and the natural frequency and strength of the lattices are presented.

A finite element model for simulating second generation high temperature superconducting coils/stacks with large number of turns

Abstract: An efficient two dimensional T-A formulation based approach is proposed to calculate the electromagnetic characteristics of tape stacks and coils made of second generation high temperature superconductors. In the approach, a thin strip approximation of the superconductor is used in which the superconducting layer is modeled as a 1-dimensional domain. The formulation is mainly based on the calculation of the current vector potential T in the superconductor layer and the calculation of the magnetic vector potential A in the whole space, which are coupled together in the model. Compared with previous T-based models, the proposed model is innovative in terms of magnetic vector potential A solving, which is achieved by using the differential method, instead of the integral method. To validate the T-A formulation model, it is used to simulate racetrack coils made of second generation high temperature superconducting (2G HTS) tape, and the results are compared with the experimentally obtained data on the AC loss. The results show that the T-A formulation is accurate and efficient in calculating 2G HTS coils, including magnetic field distribution, current density distribution, and AC loss. Finally, the proposed model is used for simulating a 2000 turn coil to demonstrate its effectiveness and efficiency in simulating large-scale 2G HTS coils.

A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian

Abstract: In Acosta etal. (2017), a complete n-dimensional finite element analysis of the homogeneous Dirichlet problem associated to a fractional Laplacian was presented. Here we provide a comprehensive and simple 2D MATLAB finite element code for such a problem. The code is accompanied with a basic discussion of the theory relevant in the context. The main program is written in about 80 lines and can be easily modified to deal with other kernels as well as with time dependent problems. The present work fills a gap by providing an input for a large number of mathematicians and scientists interested in numerical approximations of solutions of a large variety of problems involving nonlocal phenomena in two-dimensional space.