Implementation Of Finite Element Method Research Articles

On the compilation of the bathymetric data of the bay of Bengal domain in the nodal points of triangular mesh to be compatible for the implementation of finite element method

This study focuses on the compilation of bathymetric data for the Bay of Bengal domain, specifically for the area between 15°N to 23°N latitudes and 85°E to 95°E longitudes, to ensure compatibility with the Finite Element Method (FEM). A high-resolution color map of the region was generated using ArcGIS software, which served as the basis for discretizing the domain into a triangular mesh. Bathymetric data for the Bay of Bengal, including the Meghna River, was then compiled using a combination of recent data from the Bangladesh Inland Water Transport Authority (BIWTA) and generated data, which were interpolated using a two-way cubic spline interpolation method. The Modified Inverse Distance Weighted Interpolation (MIDWI) method was employed to map this bathymetric data to the nodal points of the triangular mesh, ensuring accurate representation of the seafloor’s varying depths. This work is essential for developing a reliable storm surge prediction model for the Bay of Bengal using FEM, as it provides the necessary geometric and bathymetric details required for precise numerical modeling.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces

We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An hp a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.

Experimental Characterization and Phase-Field Damage Modeling of Ductile Fracture in AISI 316L

(1) Modeling and characterization of ductile fracture in metals is still a challenging task in the field of computational mechanics. Experimental testing offers specific responses in the form of crack-mouth (CMOD) and crack-tip (CTOD) opening displacement related to applied force or crack growth. The main aim of this paper is to develop a phase-field-based Finite Element Method (FEM) implementation for modeling of ductile fracture in stainless steel. (2) A Phase-Field Damage Model (PFDM) was coupled with von Mises plasticity and a work-densities-based criterion was employed, with a threshold to propose a new relationship between critical fracture energy and critical total strain value. In addition, the threshold value of potential internal energy—which controls damage evolution—is defined from the critical fracture energy. (3) The material properties of AISI 316L steel are determined by a uniaxial tensile test and the Compact Tension (CT) specimen crack growth test. The PFDM model is validated against the experimental results obtained in the fracture toughness characterization test, with the simulation results being within 8% of the experimental measurements. (4) The novel implementation offers the possibility for better control of the ductile behavior of metallic materials and damage initiation, evolution, and propagation.

Investigation of shaded pole stator topology for low power AC motors design

PurposeThis paper aims to investigate the features of single phase shaded pole stator with squirrel–cage rotor or permanent magnet rotor, that leads to an investigation of these topoloties as induction motor or synchronous motor. The comparative analysis is realised for the following three topologies: single phase shaded pole induction motor (SPIM) with squirrel–cage rotor, the second topology (single phase synchronous motor) has the same stator configuration but with permanent magnet rotor and the third investigated topology is similar to the second one, where the stator poles instead of iron steel are made of soft composite material.Design/methodology/approachThe investigation in this work starts with a performance analysis of single-phase SPIM. Afterwards for the same stator topology the squirrel rotor is replaced with a two-pole permanent magnet rotor and the same performance analysis is realised for this topology. Finally, the second topology is improved bay replacing the iron steel stator poles with stator poles made of soft magnetic composite material and performance analysis is realised for this third type of topology as well. The performance analysis of all topologies is realised by implementation of finite element method and finite element analysis.FindingsThe presented data and diagrams from the realized investigation show that single phase synchronous motor with shaded pole stator has an improved characteristics in comparison with the initial single-phase SPIM. Finally, the third topology realized on the bases of the single-phase synchronous motor has the best performance characteristics due to the implementation of soft magnetic material in the realization of the stator poles. The proposed methodology for structural and performance improvement of a single-phase SPIM topology opens the possibility for additive manufacturing application and significant cost reduction.Originality/valueThe focus was put on exploration the possibilities of the single-phase shaded pole stator topology for application in low-power and low-cost single phase self-starting motors. By simple replacement of the squirrel–cage rotor, in the reference AKO-16 motor, with one-piece ferrite permanent magnet rotor, the self-starting single phase synchronous motor was derived. In the next step, owing to simplify the SPPM motor production process and manufacturing, the stator poles instead of iron steel lamination were made of soft composite material Somaloy®. It opens the possibility for additive manufacturing application and significant cost reduction.

Incorporation of self-heating effect into a thermo-mechanical coupled constitutive modelling for elastomeric polyurethane

Elastomeric polyurethane (EPU) is characterised by distinctive mechanical properties, including high toughness, low glass transition temperature, and high impact resistance, that render it indispensable in diverse engineering applications from soft robotics to anti-collision devices. This study presents a thermo-mechanically coupled constitutive model for EPU, systematically incorporating hyperelasticity, viscoelasticity, thermal expansion, and self-heating effect in a thermodynamically consistent manner. Experimental data, obtained from previous studies, are then used for parameter identification and model validation, including iterative updates for temperature parameters considering the self-heating effect. Subsequently, the validated model is integrated into finite element codes, i.e., user subroutine to define a material’s mechanical behaviour (UMAT) based on the commercial finite element software ABAQUS, for the computation of three-dimensional stress-strain states, facilitating the analysis of the structural response to various mechanical loads and boundary conditions. The results obtained from simulations are compared with analytical solutions to confirm the precision of Finite Element Method (FEM) implementation. The self-heating effect is further analysed under different strain rates and temperatures. To validate the engineering significance of the FEM implementation, a plate with a hole structure is also simulated. In conclusion, this research provides a robust tool for engineers and researchers working with soft materials, enhancing their understanding and predictive capabilities, notably addressing the self-heating effect in thermo-mechanical behaviours.

COMPUTATIONAL MODELING OF A FOOD PACKAGING USING FINITE ELEMENTS

Food packaging has, among others, the main function of protecting food from external biochemical contamination. Nowadays, packages can be formed by different combinations of materials in order to maximize the barrier against external agents, so ensuring food safety. This naturally leads to understand modern packages as multilayer objects, with different characteristics and purposes. In this work, it’s intended to simulate the diffusion of oxygen (responsible for the oxidation of food) in a package consisting of three polymeric materials (in ABA and ABC format) by means of a mathematical model, which resulted in a partial differential equation to be solved, along with initial and boundary conditions. Then, a Finite Element Method (FEM) implementation was devised using Fortran 77, not only to get results for the firstly chosen materials, but also to try other polymer combinations, thus generating different sets of data that could be compared further.

Homogenization-based chemomechanical properties of dissipative heterogeneous composites under transient mass diffusion

The chemomechanical properties of heterogeneous composites under mass diffusion are of significance in modern advanced technology and engineering applications. A homogenization-based two-scale chemomechanical model is developed for heterogeneous composites undergoing chemical mass diffusion. A two-scale incremental variational formulation is established for heterogeneous composites consisting of multiconstituents featuring local dissimilar diffusion-deformation properties. The minimization problems for coupled chemomechanical behaviors are solved for both macrostructure and microstructure contexts, where the macroscopic material properties are extracted from the results of local boundary value problem on the nested representative volume elements (RVEs). Through a staggered finite element method (FEM) implementation procedure, the proposed homogenization-based two-scale solution algorithm is implemented in the FEM package ABAQUS (V6.14). The developed variational model and tangential algorithm is checked by solving chemomechanical properties of particles enforced composite, where several numerical examples are conducted applying two-scale solution algorithm and validated by full-scale simulations. Parametric studies are carried out on the size effects of RVEs, with respect to the ‘inertia effect’ associated with ‘moment of mass concentration’, and the coupling mechanisms are discussed for mechanical and chemical solutions. To the end, the inelastic dissipations are solved on subscale BVPs and their effects on the mechanical deformation and chemical mass diffusion are checked. The contributions of this work are mainly two-folds. One is the theoretical advance for self-consistent homogenization modeling of the coupled multi-physics of heterogeneous composites, and a rigorous FE2 solution procedure. The other is providing numerical reference for evaluation of approximation algorithm as well as advanced data-driven method, which is needed for high-efficient material design.

Coupled Resonator Acoustic Waveguides‐Based Acoustic Interferometers Designed within 2D Phononic Crystals: Experiment and Theory

AbstractThe acoustic response of defect‐based acoustic interferometer‐like designs, known as Coupled Resonator Acoustic Waveguides (CRAWs), in 2D phononic crystals (PnCs) is reported. The PnC is composed of steel cylinders arranged in a square lattice within a water matrix with defects induced by selectively removing cylinders to create Mach‐Zehnder‐like (MZ) defect‐based interferometers. Two defect‐based acoustic interferometers of MZ‐type are fabricated, one with arms oriented horizontally and another one with arms oriented diagonally, and their transmission features are experimentally characterized using ultrasonic spectroscopy. The experimental data are compared with finite element method (FEM) simulations and with tight‐binding (TB) calculations in which each defect is treated as a resonator coupled to its neighboring ones. Significantly, the results exhibit excellent agreement indicating the reliability of the proposed approach. This comprehensive match is of paramount importance for accurately predicting and optimizing resonant modes supported by defect arrays, thus enabling the tailoring of phononic structures and defect‐based waveguides to meet specific requirements. This successful implementation of FEM and TB calculations in investigating CRAWs systems within PnCs paves the way for designing advanced acoustic devices with desired functionalities for various practical applications, demonstrating the application of solid‐state electronics principles to underwater acoustic devices description.

Fundamental data structures for matrix-free finite elements on hybrid tetrahedral grids

This paper presents efficient data structures for the implementation of matrix-free finite element methods on block-structured, hybrid tetrahedral grids. It provides a complete categorization of all geometric sub-objects that emerge from the regular refinement of the unstructured, tetrahedral coarse grid and describes efficient iteration patterns and analytical linearization functions for the mapping of coefficients to memory addresses. This foundation enables the implementation of fast, extreme-scalable, matrix-free, iterative solvers, and in particular geometric multigrid methods by design. Their application to the variable-coefficient Stokes system subject to an enriched Galerkin discretization and to the curl-curl problem discretized with Nédélec edge elements showcases the flexibility of the implementation. Finally, the solution of a curl-curl problem with 1.6 ⋅ 10 11 (more than one hundred billion) unknowns on more than 32,000 processes with a matrix-free full multigrid solver demonstrates its extreme-scalability.

Convex optimization-based structure-preserving filter for multidimensional finite element simulations

In simulation sciences, capturing the real-world problem features as accurately as possible is desirable. Methods popular for scientific simulations such as the finite element method (FEM) and finite volume method (FVM) use piecewise polynomials to approximate various characteristics of a problem, such as the concentration profile and the temperature distribution across the domain. Polynomials are prone to creating artifacts such as Gibbs oscillations while capturing a complex profile. An efficient and accurate approach must be applied to deal with such inconsistencies to obtain accurate simulations. This often entails dealing with negative values for the concentration of chemicals, exceeding a percentage value over 100, and other such problems. We consider these inconsistencies in the context of partial differential equations (PDEs). We propose an innovative filter based on convex optimization to deal with the inconsistencies observed in polynomial-based simulations. In two or three spatial dimensions, additional complexities are involved in solving the problems related to structure preservation. We present the construction and application of a structure-preserving filter with a focus on multidimensional PDEs. Methods used such as the Barycentric interpolation for polynomial evaluation at arbitrary points in the domain and an optimized root-finder to identify points of interest, improve the filter efficiency, usability, and robustness. Lastly, we present numerical experiments in 2D and 3D using discontinuous Galerkin formulation and demonstrate the filter's efficacy to preserve the desired structure. As a real-world application, implementation of the mathematical biology model involving platelet aggregation and blood coagulation has been reviewed and the issues around FEM implementation of the model are resolved by applying the proposed structure-preserving filter.

3D modeling of vector/edge finite element method for multi-ablation technique for large tumor-computational approach.

Microwave ablation (MWA) is a cancer thermal ablation treatment that uses electromagnetic waves to generate heat within the tissue. The goal of this treatment is to eliminate tumor cells while leaving healthy cells unharmed. During MWA, excess heat generation can kill healthy cells. Hence, mathematical models and numerical techniques are required to analyze the heat distribution in the tissue before the treatment. The aim of this research is to explain the implementation of the 3D vector finite element method in a wave propagation model that simulates the specific absorption rate in the liver. The 3D Nedelec elements from H(curl; Ω) space are used to discretize the wave propagation model, and this implementation is helpful in solving many real-world problems that involve electromagnetic propagation with perfect conducting and absorbing boundary conditions. One of the difficulties in ablation treatment is creating a large ablation zone for a large tumor (diameter greater than 3 cm) in a short period of time with minimum damage to the surrounding tissue. This article addresses the aforementioned issue by introducing four antennas into the different places of the tumor sequentially and producing heat uniformly over the tumor. The results demonstrated that 95.5% of the tumor cells were killed with minimal damage to the healthy cells when the heating time was increased to 4 minutes at each position. Subsequently, we studied the temperature distribution and localised tissue contraction in the tissue using the three-dimensional bio-heat equation and temperature-time dependent model, respectively. The local tissue contraction is measured at arbitrary points in the domain and is more noticeable at temperatures higher than 102°C. The thermal damage in the liver during MWA treatment is investigated using the three-state cell death model. The system of partial differential equations is solved numerically due to the complex geometry of the domain, and the results are compared with experimental data to validate the models and parameters.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation

We consider the reliable implementation of high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems. We construct a reliable algorithm to merge small interface elements with their surrounding elements to automatically generate the finite element mesh whose elements are large with respect to both domains. We propose new basis functions for the interface elements to control the growth of the condition number of the stiffness matrix in terms of the finite element approximation order, the number of elements of the mesh, and the interface deviation which quantifies the mesh resolution of the geometry of the interface. Numerical examples are presented to illustrate the competitive performance of the method.

The effect analysis of the finite element approximation order on the protective polymer layer contact deformation

Problems implementation using the finite element method (FEM) is primarily associated with the proper selection of approximation order of the shape functions, material behavior models, the coupling nature analysis and the implementation techniques choice. The choice of numerical solution parameters affects the computational procedures. The paper solves problem of the FEM parameters selection and its realization methods in order to achieve a reasonable balance between the counting time and the solution quality. Simplex elements have shape functions with linear dependences on the coordinates. The FEM implementation with the use of simplex elements has proven itself during the last decades. However, one can note the description inaccuracy of curvilinear boundaries of research objects, which can introduce additional inaccuracies in the problem numerical solution. Second-order elements will allow describing curvilinear boundaries, but require a large expenditure of computational power especially for nonlinear mechanics problems. The development of numerical application software packages allows us to evaluate the influence of the FEM approximation degree on the problem solution with the use of grid methods and computer resources at the present time. The paper found that an increase in the numerical solution error is observed at the point of initial steel - polymer contact. The study considers the elements of order 1 and 2 to analyze the influence of the FEM approximation order on the problem numerical solution. The study is carried out within the framework of the classical Hertz problem. In addition the paper investigate obtained solutions for modeling different characters of contact surface coupling for the problem of contact between a spherical die and a steel half-space through a protective polymer coating 4 mm to 12 mm thick. The study reveal that the elements order has the least influence on the problem solution at ideal contact and the greatest one at frictional contact.

On the Approximation of the Bay of Bengal Domain to be Compatible for the Implementation of Finite Element Method

In this study, the Bay of Bengal domain has been approximated using triangular mesh so that the finite element method (FEM) can be employed on it. The area between 15º N and 23º N Latitudes and 85º E and 95º E Longitudes is considered as the physical domain. A MATLAB routine and the cubic spline interpolation have been used to extract the coordinates of the points on the boundary of the whole domain and the points on the boundary of the islands from a colour image of the domain. A C++ routine is used to generate an edp file for triangular mesh using the extracted coordinates. Then FreeFem++ is used to create triangular mesh for the whole domain from the generated edp file. All the major islands are also incorporated in the final mesh. The obtained triangular mesh can be used to develop a storm surge prediction model for the Bay of Bengal region implementing FEM.

Fracture parameters analysis of stationary cracks under dynamic loads based on the field-enriched finite element method

In this paper, a field-enriched finite element method (FE-FEM) is proposed to simulate the dynamic fracture of stationary cracks in linear elastic solids. The dynamic basic theory and numerical implementation of the field-enriched finite element method are introduced in detail. Through the benchmark example, the convergence analysis of the fracture parameters is carried out. The dynamic stress intensity factors (DSIFs) considering inertia effect are calculated. The effectiveness of this strategy is first verified by two dynamic fracture examples of rectangular plates with horizontal cracks and inclined cracks, respectively. Then, the proposed method is applied to the dynamic fracture analysis of rectangular plates with holes and cracks, in which different load forms, such as step load, blasting load, cyclic load and harmonic load, are considered. In addition, the effect of the wave field on the dynamic parameters of the tip of the edge crack in plate subjected to impact loads is also investigated. The results show that the proposed numerical method has good stability and convergence, and can effectively analyze the change of fracture parameters of stationary crack under dynamic loads.

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

An eXtended finite element method implementation in COMSOL multiphysics: Thermo-hydro-mechanical modeling of fluid flow in discontinuous porous media

This paper presents the implementation of the eXtended Finite Element Method (XFEM) in the general-purpose commercial software package COMSOL Multiphysics for multi-field thermo-hydro-mechanical problems in discontinuous porous media. To this end, an exclusive enrichment strategy is proposed in compliance with the COMSOL modeling structure. COMSOL modules and physics interfaces are adopted to take account of the relevant physical processes involved in thermo-hydro-mechanical coupling analysis, namely: the mechanical deformation, fluid flow in porous media and heat transfer. Essential changes are made to the internal variables of the physics interfaces to ensure consistency in the evaluation of enriched solution fields. The model preprocessing, level-set updates, coupling of the relevant physics interfaces and postprocessing procedures are performed adopting a coherent utilization of the COMSOL’s built-in features along with the COMSOL’s LiveLink for MATLAB functions. The implementation process, remedies for the treatment of the enriched zones, XFEM framework setup, multiphysics coupling, numerical integration and numerical solution strategy are described in detail. The capabilities and performance of the proposed approach are investigated by examining several multi-field thermo-hydro-mechanical simulations involving single/multiple discontinuities in 2D/3D porous rock settings.

Parametrization-free locally-conformal perfectly matched layer method for finite element solution of Helmholtz equation

We present a novel Locally-Conformal Perfectly Matched Layer (LCPML) method for mesh truncation in the Finite Element Method (FEM), to solve wave equations derived from Maxwell's equations. The original LCPML method was proposed by Ozgun and Kuzuoglu in 2006 [16] by utilizing the concept of complex elements in which nodal coordinates of elements are replaced by complex coordinates obtained by using a complex coordinate-transformation. In the proposed LCPML method, named as LCPML-log method, a more efficient coordinate-transformation is used together with new FEM formulation and implementation. The proposed method offers the advantage that only a few (or even a single) PML layer is sufficient to get reliable results. In addition, it does not require any PML parameter tuning, and hence, the decaying wave behavior in PML is achieved self-adaptively. The numerical results of some detailed and systematic performance analyses are also presented.

On Long-Range Characteristic Length Scales of Shell Structures

Shell structures have a rich family of boundary layers including internal layers. Each layer has its own characteristic length scale, which depends on the thickness of the shell. Some of these length scales are long, something that is not commonly considered in the literature. In this work, three types of long-range layers are demonstrated over an extensive set of simulations. The observed asymptotic behavior is consistent with theoretical predictions. These layers are shown to also appear on perforated structures underlying the fact these features are properties of the elasticity equations and not dependent on effective material parameters. The simulations are performed using a high-order finite element method implementation of the Naghdi-type dimensionally reduced shell model. Additionally, the effect of the perforations on the first eigenmodes is discussed. One possible model for buckling analysis is outlined.

The method of fictitious discrete models in calculations of composite bodies

As is known, the calculation of the static strength of elastic composite bodies (CB) is reduced to finding the maximum equivalent stresses for these bodies. The finite element method (FEM) is widely used for the analysis of the stress state of CB. The basic discrete models (BM), which take into account the inhomogeneous structure of bodies in the framework of a micro-approach, have a high dimension. To reduce the dimension of discrete models, multigrid finite elements (MgFE) are effectively used. However, there are BM CB (for example, BM bodies with a micro-homogeneous structure), which have such a high dimension that the implementation of FEM for such BM using MgFE, due to limited computer resources, is difficult. To solve this problem, it is proposed to use fictitious discrete models whose dimensions are less than the dimension of the BM CB. In this paper, we propose a method of fictitious discrete models (MFDM) for calculating the strength of elastic bodies with an inhomogeneous, micro-homogeneous regular structure. The proposed method is implemented using FEM with the use of MgFE and adjusted strength conditions that take into account the error of approximate solutions. The method is based on the position that the solutions that meet the BM CB differ little from the exact ones. The calculation of CB according to MFDM is reduced to the construction and calculation of the strength of fictitious discrete models (FM), which have the following properties. The FM reflects: the shape, characteristic dimensions, attachment, loading and type of the inhomogeneous structure of the CB, and the distribution of elastic modulus corresponding to the BM CB. The FM dimension is less than the BM dimension of the CB. The sequence consisting of FM converges to BM, i.e. the limiting FM coincides with BM. The convergence of such a sequence ensures uniform convergence of the maximum equivalent voltages of the FM to the maximum equivalent voltage of the BM. Two types of FM are considered. The first type of FM consists of scaled discrete models, the second type consists of FM with variable characteristic dimensions. Calculations show that the implementation of FEM for FM using MgFE leads to a large saving of computer resources, which allows the use of MFDM for bodies with a micro-homogeneous regular structure. The calculation of the strength of CB according to MFDM requires times less computer memory than a similar calculation using BM CB, and does not contain a procedure for grinding BM. The use of adjusted strength conditions allows us to use approximate solutions with a large error in the calculations of CB for strength, which leads to an increase in the efficiency of MFDM. The given example of calculating the strength of a beam with an inhomogeneous regular fibrous structure according to MFDM shows its high efficiency.