Finite Element Toolbox Research Articles

Modeling single-phase transverse flows in a PWR rod bundle at low Reynolds number

In normal functioning conditions, the primary component of the flow within the rod bundle of a PWR core is in the axial direction, along the rods. However, in accidental situations, such as the refilling or reflooding phase during a Loss of Coolant Accident, or a Steam Line Break accident, transverse flows may significantly affect thermal-hydraulic properties in the core. Such effects have received little attention in system codes such as CATHARE, particularly in anisotropic rod structures at low Reynolds numbers. Here, we develop macroscopic pressure drop models for flows spanning from the creeping regime to unsteady vortex shedding. The averaged model is a generalized Darcy–Forchheimer equation with an apparent permeability including a rotation matrix and a non-linear dimensionless coefficient. The constitutive relations for the intrinsic permeability matrix are derived from polynomial regressions based on results obtained from microscale numerical simulations at different flow directions and Reynolds numbers. We finally test our approach in a case where transverse flow is imposed through a differential of inlet velocities and validate by comparing results from CATHARE with those obtained from a finite element toolbox.

A finite element toolbox for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present a finite element toolbox for the computation of Bogoliubov-de Gennes modes used to assess the linear stability of stationary solutions of the Gross-Pitaevskii (GP) equation. Applications concern one (single GP equation) or two-component (a system of coupled GP equations) Bose-Einstein condensates in one, two and three dimensions of space. An implementation using the free software ▪is distributed with this paper. For the computation of the GP stationary (complex or real) solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicated libraries for the associated eigenvalue problem. Mesh adaptivity is proved to considerably reduce the computational time for cases implying complex vortex states. Programs are validated through comparisons with known theoretical results for simple cases and numerical results reported in the literature. Program summaryProgram Title: FFEM_BdG_toolbox.zipCPC Library link to program files:https://doi.org/10.17632/dgypyc34gb.1Licensing provisions: Apache 2.0Programming language:▪(v 4.12) free software (www.freefem.org)Nature of problem: The software computes Bogoliubov-de Gennes (BdG) complex modes of Bose-Einstein condensates described by the Gross-Pitaevskii (GP) equation. BdG equations are obtained by linearizing the GP equation (or the system of coupled GP equations) around a stationary solution. Obtained BdG modes are used to assess on the stability of stationary states.Solution method: Stationary states of the GP equation are obtained by a Newton algorithm. Parameter space is explored using a continuation on the chemical potential. Once the stationary (complex or real) state is captured accurately, BdG modes are computed by solving the associated eigenvalue problem with the ARPACK library. Complex eigenvalues and eigenvectors are computed and stored. The wave function is discretized by P2 (piece-wise quadratic) Galerkin triangular (in 2D) or tetrahedral (in 3D) finite elements. Mesh adaptation is implemented to reduce the computational time. Examples are given for stationary states in one- and two-component Bose-Einstein condensates.

A novel homogenization method for periodic piezoelectric composites via diffused material interface

In this article, we propose a novel homogenization method for piezoelectric composites with periodic microstructures. In the proposed homogenization method, one can find the analytical expressions for the material properties of the composites in the representative volume element (RVE) via diffused material interface method. The availability of analytical expressions for the material properties in the entire domain of the RVE enables one to determine the effective homogenized material properties with the standard finite element method. As the proposed homogenization method regularizes the discontinuity in material properties across the interfaces, there is no need to explicitly track the material interfaces while implementing the finite element method for determining the effective material properties. Hence, one can implement the proposed homogenized method for any piezoelectric composites with complicated material interfaces and determine the effective material properties. In this study, we have considered the piezoelectric composites to be comprised of periodic microstructures where the RVE constitutes a matrix with an inclusion of a specific shape. We have carried out a study on the effect of the shape of inclusion viz. square-shaped, I-shaped, T-shaped, and plus-shaped, and the size of inclusion on the effective piezoelectric material properties. We have also studied the influence of shape and size of inclusion on the electro-mechanical response of a homogenized piezoelectric continuum. To implement the proposed homogenization method, we have used Gridap, an open-source finite element toolbox in Julia that provides very compact codes freely available to all the researchers and makes a third-party verification of the proposed method straightforward.

A phase-field model for electro-mechanical fracture with an open-source implementation of it using Gridap in Julia

In this paper, we propose a phase-field model for electro-mechanical fracture in brittle materials and provide an open-source implementation of the proposed model using a newly developed finite element (FE) toolbox, Gridap in Julia. Here, we have considered electric potential as an additional kinematic descriptor along with the displacement and phase-field variable. Using the virtual power principle, we have derived force balances for the electro-mechanical forces and a force balance associated with the damage. To incorporate the strong electro-mechanical coupling effect into the model, we have considered the Helmholtz free energy as a function of the electric field vector, strain tensor, phase-field variable, and the gradient of the phase-field variable. The proposed model ensures that the constitutive relations of the thermodynamic fluxes are on the satisfaction of the thermodynamic laws and can readily accommodate dissipative energy effects whenever needed. The proposed model provides complex crack paths as a solution to the governing partial differential equations (PDEs), and thus bypasses the need for re-meshing, enrichment of FE shape functions, and an explicit tracking of the crack surfaces. The use of Gridap makes the FE implementation of the proposed model exceedingly compact and user-friendly requiring very low memory usage and providing a high degree of flexibility to the users in defining weak forms of the governing PDEs. We have validated the proposed model and its implementation against a compact tension test on a single edge notched plate and a set of three-point bending tests on a notched beam made of lead zirconium titanate (PZT)-4 piezoelectric ceramics.

Combined diffused material interface and hybrid phase-field model for brittle fracture in heterogeneous composites

In this article, we propose a novel approach for modeling brittle fracture in heterogeneous composites using a combined diffused material interface method and a hybrid phase-field model (PFM). We have used the diffused material interface method to derive analytical expressions for the regularized material properties across the material interfaces that are treated as an input to the PFM for the simulation of brittle fractures in composites. The use of diffused material interface method regularizes the material properties in such a way that one can avoid the discontinuity in stress across the material interface and thus bypass the requirement of explicit tracking of material interfaces for finite element implementation of the hybrid PFM. We have implemented the proposed combined model using a recently developed open-source finite element toolbox, Gridap in Julia. We have considered tests on a single circular fiber inclusion under tension, a bi-material rectangular plate under tension, single square inclusion under tensile loading, a notched composite beam with a central notch under symmetric three-point bending, and a rectangular plate with multiple fibers under tensile loading to show how the proposed combined model can be used to simulate brittle fracture in composites.

A phase-field model for thermo-mechanical fracture

In this article, we propose a thermodynamically consistent phase-field model for thermo-mechanical fracture and provide an open-source implementation of the proposed model using a recently developed finite element toolbox, Gridap in Julia. Here, we have derived the balance equations for the thermo-mechanical fracture by invoking the virtual power principle and determined the constitutive relations for the thermodynamic fluxes based on the satisfaction of the thermodynamic laws. Our proposed formulation provides an equation of temperature evolution that can easily accommodate dissipative effects such as viscous damping. We provide very compact and user-friendly open-source codes for implementing the proposed model using Gridap in Julia that requires very low memory usage and gives a high degree of flexibility to the users in defining weak forms of the governing partial differential equations (PDEs). We have validated the proposed model and its implementation against such standard results available in the literature as crack propagation in the cruciform shape material, single edge notched plate, bi-material beam, and a quenching test.

High-order Numerical Quadratures in a Tetrahedron with an Implicitly Defined Curved Interface

Given a shape regular tetrahedron and a curved surface that is defined implicitly by a nonlinear level set function and divides the tetrahedron into two sub-domains, a general-purpose, robust, and high-order numerical algorithm is proposed in this article for computing both volume integrals in the sub-domains and surface integrals on their common boundary. The algorithm uses a direct approach that decomposes 3D volume integrals or 2D surface integrals into multiple 1D integrals and computes the 1D integrals with Gaussian quadratures. It only requires finding roots of univariate nonlinear functions in given intervals and evaluating the integrand, the level set function, and the gradient of the level set function at given points. It can achieve arbitrarily high accuracy by increasing the orders of Gaussian quadratures, and it does not need extra a priori knowledge about the integrand and the level set function. The code for the algorithm is freely available in the open-source finite element toolbox Parallel Hierarchical Grid (PHG) and can serve as a basic building block for implementing 3D high-order numerical algorithms involving implicit interfaces or boundaries.

Kaskade 7 — A flexible finite element toolbox

Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables.We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way.

Blast-load-induced interaction between adjacent multi-story buildings

The present study aims to present a comprehensive understanding of the performance of neighboring multi-story buildings with different dynamic characteristics under blast loads. Two different scenarios are simulated in terms of explosion locations with respect to both buildings. To investigate the effect of interaction between the neighboring buildings in terms of the induced responses, the separation gap is set to be sufficiently small to ensure collisions between stories. An adequately large separation gap is set between the buildings to explore responses without collisions under the applied blast loads. Several blast loads with different peak pressure intensities are employed to perform the dynamic analysis. The finite-element toolbox Computer Aided Learning of the Finite-Element Method (CALFEM) is used to develop a MATLAB code to perform the simulation analysis. The dynamic responses obtained in the scenarios considered herein are presented comparatively. It is found that the obtained stories\' responses are governed mainly by the location and intensity of the applied blast loads, separation distances, and flexibility of the attacked structures. Moreover, explosions near a light and flexible building may lead to a significant decrease in blast resistance because explosions severely influence the dynamic responses of the building\'s stories.

CastLab: an object-oriented finite element toolbox within the Matlab environment for educational and research purposes in computational solid mechanics

The Matlab environment has become widely used among the computational mechanics community, not only for research purposes but also to teach either undergraduate or graduate classes. This paper aims to present a new toolbox devoted to computational mechanics and in particular to solid mechanics. Both recent and well-established numerical formulations have been implemented in it. One of its strengths resides in the fact that it was developed within an object-oriented framework. This key feature makes the CastLab toolbox easy-to-use and with extensive capabilities for customized user developments. After a brief description of the theoretical background related to the problems that can be solved by means of the toolbox, several representative case-studies are presented. These examples have been selected to illustrate not only the numerical efficiency of the toolbox, which is of primary importance for research purposes, but also its strong educational and pedagogic potential.

Design optimisation for cold-formed steel residential roof truss using genetic algorithm

PurposeThis paper aims to present the design optimisation using genetic algorithm (GA) to achieve the highest strength to weight (S/W) ratio, for cold-formed steel residential roof truss.Design/methodology/approachThe GA developed in this research simultaneously optimises roof pitch, truss configurations, joint coordinates and applied loading of typical dual-pitched symmetrical residential roof truss. The residential roof truss was considered with incremental uniform distributed loading, in both gravitational and uplift directions. The structural analyses of trusses were executed in this GA using finite element toolbox. The ultimate strength and serviceability of trusses were checked through the design formulation implemented in GA, according to the Australian standard, AS/NZS 4600 Cold-formed Steel Structures.FindingsAn optimum double-Fink roof truss which possess highest S/W ratio using GA was determined, with optimum roof pitch of 15°. The optimised roof truss is suitable for industrial application with its higher S/W ratio and cost-effectiveness. The combined methodology of multi-level optimisation and simultaneous optimisation developed in this research could determine optimum roof truss with consistent S/W ratio, although with huge GA search space.Research limitations/implicationsThe sizing of roof truss member is not optimised in this paper. Only single type of cold-formed steel section is used throughout the whole optimisation. The design of truss connection is not considered in this paper. The corresponding connection costs are not included in the proposed optimisation.Practical implicationsThe optimum roof truss presented in this paper is suitable for industrial application with higher S/W ratio and lower cost, in either gravitational or uplift loading configurations.Originality/valueThis research demonstrates the approaches in combining multi-level optimisation and simultaneous optimisation to handle large number of variables and hence executed an efficient design optimisation. The GA designed in this research determines the optimum residential roof truss with highest S/W ratio, instead of lightest truss weight in previous studies.

FELICITY: A Matlab/C++ Toolbox for Developing Finite Element Methods and Simulation Modeling

This paper describes a MATLAB/C++ finite element toolbox, called FELICITY, for simulating various types of systems of partial differential equations (e.g., coupled elliptic/parabolic problems) usin...

Dynamic Response of Adjacent Buildings Under Explosive Loads

The aim of this paper was to investigate the effect of an explosive load on the dynamic response of adjacent buildings including interaction due to collisions. Two buildings with different dynamic characteristics, to ensure one building behaves as a flexible and the neighboring one behaves as a stiff, are selected for the investigation. For such analysis, two different cases of adjacent buildings are modeled: (1) no interaction between adjacent buildings, sufficient separation gap exists, and (2) interaction between adjacent buildings, insufficient separation gap exists. The nonlinear viscoelastic model is employed to capture the induced force due to collisions. The influence of variations of a number of factors such as (a) separation gaps, (b) natural periods, and (c) peak over static pressures is considered. The response of buildings is studied under blast loads due to a source of explosion near the flexible building. The analysis is repeated using a source of explosion near the heavy and stiff one. Numerical simulations for estimating the dynamic response such as adjacent building structures are carried out using a developed MATLAB code based on the finite element toolbox CALFEM. Displacements and pounding forces due to collisions are determined. The displacements obtained from the analyses of adjacent buildings with insufficient separation gap are compared with those obtained assuming sufficiently separated buildings. It is found that explosive loads can cause excessive displacements leading to collisions between insufficiently separated buildings. Such collisions significantly affect the induced dynamic response of the adjacent two buildings, especially the flexible one.

FEMAG: A High Performance Parallel Finite Element Toolbox for Electromagnetic Computations

This paper presents a parallel finite element toolbox for computing large electromagnetic devices on unstructured tetrahedral meshes, FEMAG—Fem for ElectroMagnetics on Adaptive Grids. The finite element toolbox deals with unstructured tetrahedral meshes and can solve electromagnetic eddy current problems in both frequency domain and time domain. It adopts high-order edge element methods and refines the mesh adaptively based on reliable and efficient finite element a posteriori error estimates. We demonstrate the competitive performance of FEMAG by extensive numerical experiments, including TEAM (Testing Electromagnetic Analysis Methods) Problem 21 and the simulation for a single-phase power transformer.

Parallel multigrid method for finite element simulations of complex flow problems on locally refined meshes

We present a parallel multigrid solver on locally refined meshes for solving very complex three-dimensional flow problems. Besides describing the parallel implementation in detail, we prove the smoothing property of the suggested iteration for a simple model problem. For demonstration of the efficiency and feasibility of the solver, we show a chemically reactive flow simulation for a Methane burner using detailed chemical reaction modeling. Further, we give the results of an ocean flow simulation. All described methods are implemented in the finite element toolbox Gascoigne. Copyright © 2010 John Wiley & Sons, Ltd.

Structural mechanics approach for Carbon Nanotubes

The mechanical behavior of Carbon Nanotubes (CNT) is analyzed using the finite element modeling, differential quadrature method and verified through symbolic processing. The mechanical behavior especially the buckling and vibration problems are analyzed in this paper. The molecular mechanics approach used in earlier studies was tedious to work upon as it involves the determination of the force constants like Van der Waal forces of molecules. This necessitated the use of the finite element approach. The most popular computer package MATLAB is used as a matrix calculator for performing finite element analysis. The Finite Element Toolbox in MATLAB having built-in subroutines enables us to solve the problems quite easily. The symbolic processing tool “Mathematica” is used to verify the results. These systems are “expert systems” having remarkable capability of manipulating not only numbers, but also abstract symbols which represent the numerical quantities. The single walled Nanotubes and the double walled Nanotubes are both analyzed for static, buckling and vibration. CNTs are not perfectly straight in configuration so the imperfection analysis is also carried out. The same concept is then extended to analyze the mechanical behavior of multi-walled Nanotubes. A Matlab program is developed to analyze CNTs of any number of walls.

A parallel adaptive finite-element package based on ALBERTA

ALBERTA, a sequential adaptive finite-element toolbox, is being used widely in the fields of scientific and engineering computation, especially in the numerical simulation of electromagnetics. But the nature of sequentiality has become the bottle-neck while solving large scale problems. So we develop a parallel adaptive finite-element package based on ALBERTA, using ParMETIS and PETSc. The package is able to deal with any problem that ALBERT solved. Furthermore, it is suitable for distributed memory parallel computers including PC clusters. In this paper, we present the implementation of the package in detail, and address several key algorithms and strategies of parallelization. Finally, some numerical experiments are given to show the performance and scalability of our package.

MATLAB guide to finite elements: an interactive approach

This successful textbook explores the numerical implementation of Finite Element Analysis using the computer program MATLAB, which is very popular today in engineering and engineering education. The book contains a short tutorial on MATLAB as well as a systematic strategy for the treatment of finite element methods. Useful to both students and researchers in engineering, it provides various examples and exercises from mechanical, civil and aerospace engineering, as well as from materials science. The book especially stresses the interactive use of MATLAB, with each example solved in an interactive manner. The accompanying CD-ROM includes more than fifty MATLAB functions specifically written to be used with this book in the form of a MATLAB Finite Element Toolbox. An extensive solutions manual is provided as well, which includes detailed solutions to all the problems in the book for classroom use. This second edition includes a new brick (solid) element with eight nodes and a one-dimensional fluid flow element. A review of the applications of finite elements in various fields such as fluid flow, heat transfer, structural dynamics, electro-magnetics, is added as well.

Abstract data structures for a finite element package.: Design principles of ALBERT

AbstractAlbert is an Adaptive multi‐Level finite element toolbox using Bisectioning refinement and Error control by Residual Techniques. Its design is based on appropriate data structures holding geometrical, finite element, and algebraic information. Using such data structures, abstract adaptive methods for stationary and tame dependent problems, assembly tools for discrete systems, and dimension dependent tasks like mesh modifications can be provided in a library. This allows dimension‐independent development and programming of a general class of applications. In ALBERT, hierarchical 2d and 3d meshes are stored in binary trees. Several sets of finite elements can be used on the same mesh, either using predefined ones, or by adding new sets for special applications. Depending on the currently used finite element spaces, all degrees of freedom are automatically managed during mesh modifications.

Finite element toolbox for generic coupling (magnetic, thermal, etc.)

Software already allows one to carry out multiple coupling, however, they are often destined to only one type of coupling and it can prove difficult to integrate a new physical field. We have developed an adapted finite element toolbox for generic coupling. Thanks to an object oriented approach, it is possible to carry out multiple coupling. It is easy to add a new coupling transparently with a minimum of development. With the choice of an indirect coupling method, one can integrate specific solution methods, independently developed. We present the implementation in SIC (Systeme Interactif de Conception) software, one of the first finite element software using an object approach. We have chosen it for its FORTRAN object manager and the various physical models included. We have validated all our developments on the physical process of induction heating.