Finite Element Method Approximation Research Articles

Plane crack problems within strain gradient elasticity and mixed finite element implementation

An alternative approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C1 and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem’s initial data are formulated for discrete equations of mixed FEM. These are solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect.

Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic problems

Abstract In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks (NNs). The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and NNs act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias/approximation properties of NNs. We derive a priori error estimates in the standard $L^2(\varOmega )$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.

Efficient Newton‐multigrid FEM solver for multifield nonlinear coupled problems applied to thixoviscoplastic flows

AbstractThis note is concerned with efficient Newton‐multigrid finite element method (FEM) solver for multifield nonlinear flow problems. In our approach, for efficient FEM solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to obtain the numerical solutions. We concretize our ideas on thixoviscoplastic flow problems. It is a two‐field coupled nonlinear problem. And beside the integrated nonlinearity within momentum and microstructure equations, thixoviscoplastic problems induce a nonlinear two‐way coupling. As far as FEM numerical solutions are concerned, we set the problem in a suitable variational form to use the corresponding wellposedness analysis to develop FEM techniques for the solver. Indeed, the wellposedness study is not an intellectual exercise, rather it is the foundation for the approximate thixoviscoplastic problem as well as for the development of an efficient solver. We base our investigations for the solver on our wellposedness and error analysis results of thixoviscoplastic flow problems published in Proc. Appl. Math. Mech. We continue our series, and proceed to develop a monolithic Newton‐multigrid thixoviscoplastic solver. The solver is based on Newton's method and geometric multigrid techniques to treat the coupling of the problem. So, we use Local Pressure Schur Complement (LPSC) concept to solve the coupled problem on mesh's elements, and proceed with outer blocks Gauss‐Seidel iteration to update the global solutions. Furthermore, we handle the nonlinearity of the problem with the combined adaptive discrete Newton's and multigrid methods. The adaptivity within discrete Newton's method is based on the adaptive step‐length control for the discrete differencing in the Jacobian calculations, while the convergence of linear multigrid solver is made to match the convergence requirement of nonlinear solver, accordingly. And the solver's update parameters are solely dependent on the actual convergence rate of the nonlinear problem. We provide the numerical results of solver performance for thixoviscoplastic lid‐driven cavity flow.

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.

Well-conditioned and optimally convergent second-order Generalized/eXtended FEM formulations for linear elastic fracture mechanics

The Generalized/eXtended Finite Element Method (G/XFEM) has been established as an approach to provide optimally convergent solutions for classes of problems that are challenging for the standard version of the Finite Element Method (FEM). For problems with non-smooth solutions, as those within the Linear Elastic Fracture Mechanics (LEFM) context, the FEM convergence rates are often not optimal and are bounded by the strength of the singularity in the analytical solution. This can be overcome by G/XFEM and many researches have focused on delivering first-order convergent solutions for LEFM problems. The difficulty in obtaining higher-order accurate approximations, however, relies mainly on also controlling the growth rate of the stiffness matrix condition number. In this paper, well-conditioned and optimally convergent second-order G/XFEMs are proposed for LEFM simulations by augmenting second-order Lagrangian FEM approximation spaces. More specifically, two strategies are proposed in order to accurately represent second-order discontinuous functions along a crack. Also, a third strategy that essentially improves the use of linear Heaviside functions in the sense that these enrichments no longer cause linear dependencies among the set of G/XFEM shape functions is proposed. The numerical experiments show the robustness of the formulations presented herein for a set of crack topologies.

Transient Analysis of Micro/Nano Plates by Moving Finite Element Method

The paper deals with transient analysis of homogeneous as well as FGM (functionally graded material) thin micro/nano plates subjected to transversal dynamic loading. within the highergrade continuum theory of elasticity. The microscopic structure of material is reflected in this higher-grade continuum theory via one material coefficient called the micro-length scale parameter. Furthermore the material can be composed of two micro-constituents what is included in the employed continuum model by functional gradation of the Young’s modulus through the plate thickness with assuming power-law dependence of volume fractions of micro-constituents on the transversal coordinate. The high order derivatives of field variables are eliminated by decomposing the original governing partial differential equations (PDE) into the system of PDEs with lower order derivatives. For the numerical implementation, the weak formulation is proposed with novel Moving Finite Element approximation method of spatial variations of field variables. The semi-discretized equations of motion yield a system of ordinary differential equations which can be solved by standard time stepping techniques. Several numerical simulations are devoted to study the influence of micro-length scale parameter as well as the parameters of gradation of Young’s modulus on coupled bending and in-plane deformation response modes.

Application of finite element method to General Dynamic Equation of Aerosols – Comparison with classical numerical approximations

This paper focuses on the numerical approximation of the general dynamic equation of aerosols (GDE), an integro-partial differential equation, which models the temporal evolution of aerosol number distributions. More specifically, we study the applicability of the finite element method (FEM) to numerically approximate the GDE at different conditions. In addition to applying the conventional Galerkin FEM to the GDE, we also introduce its extension, the Petrov–Galerkin FEM to improve the stability of the approximation. The FEM approximations are compared against the so-called sectional method, based on the finite difference approximation, which is commonly used for the numerical approximation of the GDE. The FEM schemes and the sectional method are validated with series of numerical simulations, where an accurate solution of GDE is available. In these simulations, we consider special cases of pure condensation and coagulation, as well as cases including both of these processes. The accurate solutions – to which numerical approximations are compared – consist of either the analytical solutions or an accurate solution to the discrete GDE, where the particle size range is discretized into intervals of the size of a monomer, and the condensation process is modeled as an interaction between aerosol particles and monomers. The results of this paper demonstrate that while in the solution of the coagulation problem, the conventional sectional method is more efficient than the FEM, in cases of the condensation equation and the condensation-dominated GDE, the FE approximations outperform the sectional method – yielding a desired level of accuracy of the solution using less discretization points and in computation time which can be even orders of magnitude lower than with the sectional method.

Moored ship motion under the resonance conditions with breakwaters: A coupled numerical approach

A hybrid element model has been formulated to understand the wave interaction with moored ship motion using mild slope equation (MSE) including the influence of breakwaters and wave absorbers, under resonance conditions for variable bathymetry. In a hybrid element formulation, a combined approach with the finite element method (FEM) in an interior domain (port region) including moored ship region and an analytical approximation in the unbounded region (open sea) is utilized to determine the wave response of moored ship motion. Especially, in this approach, a FEM and analytical approximation are firstly utilized to solve MSE with the consideration of permeable breakwaters for moored ship motion under the resonance conditions. The non-uniform discretization pattern enhanced the numerical accuracy of the coupled numerical approach and is verified by using convergence analysis. Further, the simulation results are compared with existing studies to validate the applicability of the present numerical model. Wave absorbers and permeable breakwaters are employed to determine the incident wave response over moored ship motion inside the realistic Paradip port, Odisha, India. In addition, the efficacy of permeable breakwaters and wave absorbers on moored ship motion is investigated. It is shown that breakwaters and wave absorbers at appropriate locations play a significant role to induce/reduce the resonance inside the port. Therefore, the current numerical model provides the countermeasures tactics such as installing breakwaters, wave absorbers to mitigate the resonance in arbitrarily shaped ports for safe moorings.

Application of the generalized finite-element method to the acoustic wave simulation in exploration seismology

The generalized finite-element method (GFEM) has been applied frequently to solve harmonic wave equations, but its use in the simulation of transient wave propagation is still limited. We have applied GFEM for the simulation of the acoustic wave equation in models relevant to exploration seismology. We also perform an assessment of its accuracy and efficiency. The main advantage of GFEM is that it provides an enhanced solution accuracy compared to the standard finite-element method (FEM). This is attained by adding user-defined enrichment functions to standard FEM approximations. For the acoustic wave equation, we consider plane waves oriented in different directions as the enrichments, whose argument includes the largest wavenumber of the wavefield. We combine GFEM with an unconditionally stable time integration scheme with a constant time step. To assess the accuracy and efficiency of GFEM, we compare the GFEM simulation results against those obtained with the spectral-element method (SEM). We use SEM because it is the method of choice for wave propagation simulation due to its proven accuracy and efficiency. In the numerical examples, we first perform a convergence study in space and time, evaluating the accuracy of both methods against a semianalytical solution. Then, we consider simulations of relevant models in exploration seismology that include low-velocity features, an inclusion with a complex geometric boundary and topography. Results using these models indicate that GFEM presents comparable accuracy and efficiency to those based on SEM. For the given examples, the GFEM efficiency stems from the combined effect of local mesh refinement, nonconforming or unstructured, and the unconditionally stable time integration scheme with a constant time step. Moreover, these features provide great flexibility to the GFEM implementations, proving advantageous when using, for example, unstructured grids that impose severe time step size restrictions in SEM.

Experimental and numerical investigations on dynamic characteristics of gear-roller-bearing system

In the present study, dynamic characteristics of gear-roller-bearing system is investigated by use of experimental test and theoretical modeling. In terms of experiment, a new test set-up dedicated to offset press is developed to measure the acceleration within a wide range of speed. After the completion of data acquisition, the vibration signals obtained from different conditions are analyzed both in time domain and frequency domain, thereby capturing the source of vibration excitation. For theoretical modeling, a novel dynamic model of gear-roller-bearing system including time-varying properties and contact nonlinearities is proposed based on the fundamental idea of finite element method, divided into the helical gear pair model, dynamic model of roller and shaft, bearing elements, shoulder iron and rubber contact elements. The unconditionally stable algorithm, known as Newmark-β method, is employed to solve the governing equations, and then the validity of mathematical model in predicting dynamic response is demonstrated by comparison with experimental results. Experimental and numerical results indicate that the proposed method can be utilized as a useful tool for parameter design and performance analysis of gear-roller-bearing system.

Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity

SummaryThis paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic body is almost incompressible. Several alternative a posteriori error estimators for the energy norm of the finite element error are introduced and analysed. We establish upper and lower bounds for the energy error in terms of these error estimators and show that the constants in the bounds are independent of the Lamé coefficients: The proposed estimators are robust in the incompressible limit. We also consider the requirement for pressure stabilisation when using lowest‐order conforming approximation. Computational results are presented that validate the theoretical estimates. The software used to generate these results is available online.

A weak Galerkin finite element method for solving nonlinear convection-diffusion problems in two dimensions

We study weak Galerkin (WG) finite element method (FEM) for solving nonlinear convection-diffusion problems. A WG finite element scheme is presented based on a new variational form. We prove the energy conservation law and stability of the continuous time WG FEM. In particular, optimal order error estimates are established for the WG FEM approximation in both a discrete H1-norm and L2-norm. Numerical experiments are performed to confirm the theoretical results.

Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

The implementation of spectral stochastic Galerkin finite element approximation methods for Helmholtz equations with random inputs is considered. The corresponding linear systems have matrices represented as Kronecker products. The sparsity of such matrices is analysed and a mean-based preconditioner is constructed. Numerical examples show the efficiency of the mean-based preconditioners for stochastic Helmholtz problems, which are not too close to a resonant frequency.

Effect of Nonstructural Masses on Civil Structures by CUF-Based Finite Element Models

This paper proposes an application of already established higher-order models and, namely, investigates the free vibration analysis of civil engineering structures subjected to nonstructural masses. The refined one-dimensional theories adopted are based on the Carrera unified formulation (CUF). The stiffness and mass matrices are obtained by means of the principle of virtual displacements in conjunction with the finite element method (FEM). According to CUF, in fact, their formulation is performed in terms of fundamental nuclei, which depend neither on the adopted class of beam theory nor on the FEM approximation along the axis. In order to account for nonstructural localised masses, the fundamental nucleus of the mass matrix has been opportunely modified. In this work, Lagrange polynomials have been employed in the framework of CUF to develop pure translational displacement-based refined beam models. The models obtained using this approach are referred, in the literature, to as component wise (CW), which allows to model each structural component as a 1D element. The free vibration analysis has been carried out for different cases of nonstructural mass distribution in typical civil buildings. The results obtained are compared with classical Solid/Shell FEM solutions from the commercial code MSC Nastran. The capabilities of the CW models are demonstrated, since this formulation is able to replicate the 3D solid results with enhanced performances in terms of computational efforts. Moreover, the importance of taking into account the correct distribution of the nonstructural masses is shown to be of fundamental importance in vibration analysis of civil buildings.

A computational framework for G/XFEM material nonlinear analysis

The Generalized/eXtended Finite Element Method (G/XFEM) has been developed with the purpose of overcoming some limitations inherent to the Finite Element Method (FEM). Different kinds of functions can be used to enrich the original FEM approximation, building a solution specially tailored to problem. Certain obstacles related to the nonlinear analysis can be mitigated with the use of such strategy and the damage and plasticity fronts can be precisely represented. A FEM computational environment has been previously enclosed the G/XFEM formulation to linear analysis with minimum impact in the code structure and with requirements for extensibility and robustness. An expansion of the G/XFEM implementation to physically nonlinear analysis under the approach of an Unified Framework for constitutive models based on elastic degradation is firstly presented here. The flexibility of the proposed framework is illustrated by several examples with different constitutive models, enrichment functions and analysis models.

A study of small sized quartz tuning fork using finite element analysis

This is a preliminary study to small sized quartz tuning fork operating at 32.768 kHz for several types of supporting structure. Tine length and tine width can be calculated by Euler-Bernoulli equation (Exact Solution), but the base shall be analyzed by FEM (Approximation Solution) due to flexibility of base design. Finite Element Method (FEM) software ANSYS is used to simulate quartz tuning fork to extract electric response from harmonic analysis. Several types of supporting structure are to compare to modified tuning fork with notch by simulation.

Distributed parameter system identification using finite element differential neural networks

Most of the previous work on identification involves systems described by ordinary differential equations (ODEs). Many industrial processes and physical phenomena, however, should be modeled using partial differential equations (PDEs) which offer both spatial and temporal distributions that are simply not available with ODE models. Systems described by a PDE belong to a class of system called distributed parameter system (DPS). This article presents a method for solving the problem of identification of uncertain DPSs using a differential neural network (DNN). The DPS, assumed to be described by a PDE, is approximated using the finite element method (FEM). The FEM discretizes the domain into a set of distributed and connected nodes, thereby, allowing a representation of the DPS in a finite number of ODEs. The proposed DNN follows the same interconnection structure of the FEM, thus allowing the DNN to identify the FEM approximation of the DPS in both 2D and 3D domains. Lyapunov's second method was used to derive adaptive learning laws for the proposed DNN structure. The identification algorithm, here developed in Nvidia's CUDA/C to reduce the execution time, runs mostly on the graphics processing unit (GPU). A physical experiment served to validate the 2D case. In the experiment, the DNN followed the trajectory of 57 markers that were placed on an undulating square piece of silk. The proposed DNN is compared against a method based on principal component analysis and an artificial neural network trained with group search optimization. In addition to the 2D case, a simulation validated the 3D case, where input data for the DNN was generated by solving a PDE with appropriate initial and boundary conditions over an unitary domain. Results show that the proposed FEM-based DNN approximates the dynamic behavior of both a real 2D and a simulated 3D system.

Dynamic response of aerospace structures by means of refined beam theories

The present paper is devoted to the investigation of the dynamic response of typical aerospace structures subjected to different time-dependent loads. These analyses have been performed using the mode superposition method combined with refined one-dimensional models, which have been developed in the framework of the Carrera Unified Formulation (CUF). The Finite Element Method (FEM) and the principle of virtual displacements are used to compute the stiffness and mass matrices of these models. Using CUF, one has the great advantage to obtain these matrices in terms of fundamental nuclei, which depend neither on the adopted class of beam theory nor on the FEM approximation along the beam axis. In this paper, Taylor-like expansions (TE), Chebyshev expansion (CE) and Lagrange expansion (LE) have been employed in the framework of CUF. In particular, the latter class of polynomials has been used to develop pure translational displacement-based refined beam models, which are referred to as Component Wise (CW). This approach allows to model each structural component as a 1D element. The dynamic response analysis has been carried out for several aerospace structures, including thin-walled, open section and reinforced thin-shells. The capabilities of the proposed models are demonstrated, since this formulation allows to detect shell-like behavior with enhanced performances in terms of computational efforts.

Free vibration analysis of civil engineering structures by component-wise models

Abstract Higher-order beam models are used in this paper to carry out free vibration analysis of civil engineering structures. Refined kinematic fields are developed using the Carrera Unified Formulation (CUF), which allows for the implementation of any-order theory without the need for ad hoc formulations. The principle of virtual displacements in conjunction with the finite element method (FEM) is used to formulate stiffness and mass matrices in terms of fundamental nuclei. The nuclei depend neither on the adopted class of beam theory nor on the FEM approximation along the beam axis. This paper focuses on a particular class of CUF models that makes use of Lagrange polynomials to discretize cross-sectional displacement variables. This class of models are referred to as component-wise (CW) in recent works. According to the CW approach, each structural component (e.g. columns, walls, frame members, and floors) can be modeled by means of the same 1D formulation. A number of typical civil engineering structures (e.g. simple beams, arches, truss structures, and complete industrial and civil buildings) are analyzed and CW results are compared to classical beam theories (Euler–Bernoulli and Timoshenko), refined beam models based on Taylor-like expansions of the displacements on the cross-section, and classical solid/shell FEM solutions from the commercial code MSC Nastran. The results highlight the enhanced capabilities of the proposed formulation. It is in fact demonstrated that CW models are able to replicate 3D solid results with very low computational efforts.

Modeling the pull-in behavior of electrostatically actuated micro beams by an approximate finite element method

We establish the deflection functions of electrostatically actuated micro beams by an approximate finite element method (AFEM), in which the beam and the electrostatic load are discretized. The beam is replaced with a series of beam elements by traditional FEM. Using the total differential of the distributed electrostatic force, we represent such a force with the nodal forces on the beam elements. We calculate the deformation curve of the beam by gradually loading voltage in small increments, and pull-in behavior is identified when the convergence of the deflection iteration cannot be achieved after voltage increment. The proposed method considers the effect of deformation on stiffness by establishing a new equivalent stiffness matrix for each voltage step based on of the results of previous steps. Through this approach, we prevent the approximate errors of the stiffness matrix from accumulating. The AFEM results on micro beams with different geometries indicate good agreement with those obtained by other studies and those derived using commercial FE software. Copyright © 2013 John Wiley & Sons, Ltd.