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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Simulation of metallic powder bed additive manufacturing processes with the finite element method: A critical review

Abstract: This article provides a literature review of finite element simulation studies for metallic powder bed additive manufacturing processes. The various approaches in the numerical modeling of the processes and the selection of materials properties are presented in detail. Simulation results are categorized according to three major findings’ groups (i.e. temperature field, residual stresses and melt pool characteristics). Moreover, the means used for the experimental validation of the simulation findings are described. Looking deeper into the studies reviewed, a number of future directions are identified in the context of transforming simulation into a powerful tool for the industrial application of additive manufacturing. Smart modeling approaches should be developed, materials and their properties should be further characterized and standardized, commercial packages specialized in additive manufacturing simulation have to be developed and simulation needs to become part of the modern digital production chai...

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.

Simplified Damage Plasticity Model for Concrete

Abstract: The past several years have witnessed an increase in research on the nonlinear analysis of the structures made from reinforced concrete. Several mathematical models were created to analyze the beha...

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.

In-situ X-ray computed tomography characterisation of 3D fracture evolution and image-based numerical homogenisation of concrete

Abstract: In-situ micro X-ray Computed Tomography (XCT) tests of concrete cubes under progressive compressive loading were carried out to study 3D fracture evolution. Both direct segmentation of the tomography and digital volume correlation (DVC) mapping of the displacement field were used to characterise the fracture evolution. Realistic XCT-image based finite element (FE) models under periodic boundaries were built for asymptotic homogenisation of elastic properties of the concrete cube with Young's moduli of cement and aggregates measured by micro-indentation tests. It is found that the elastic moduli obtained from the DVC analysis and the FE homogenisation are comparable and both within the Reuss-Voigt theoretical bounds, and these advanced techniques (in-situ XCT, DVC, micro-indentation and image-based simulations) offer highly-accurate, complementary functionalities for both qualitative understanding of complex 3D damage and fracture evolution and quantitative evaluation of key material properties of concrete.

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.

Investigation of Gallium Nitride Devices in High-Frequency LLC Resonant Converters

Abstract: Newly emerged gallium nitride (GaN) devices feature ultrafast switching speed and low on-state resistance that potentially provide significant improvements for power converters. This paper investigates the benefits of GaN devices in an LLC resonant converter and quantitatively evaluates GaN devices’ capabilities to improve converter efficiency. First, the relationship of device and converter design parameters to the device loss is established based on an analytical model of LLC resonant converter operating at the resonance. Due to the low effective output capacitance of GaN devices, the GaN-based design demonstrates about 50% device loss reduction compared with the Si-based design. Second, a new perspective on the extra transformer winding loss due to the asymmetrical primary-side and secondary-side current is proposed. The device and design parameters are tied to the winding loss based on the winding loss model in the finite element analysis (FEA) simulation. Compared with the Si-based design, the winding loss is reduced by 18% in the GaN-based design. Finally, in order to verify the GaN device benefits experimentally, 400- to 12-V, 300-W, 1-MHz GaN-based and Si-based LLC resonant converter prototypes are built and tested. One percent efficiency improvement, which is 24.8% loss reduction, is achieved in the GaN-based converter.

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.