Finite Element Solution Research Articles

Subscale modeling of material flow in orthogonal metal cutting

Enhanced simulation capability for the cutting process is crucial to making quick evaluations of cutting forces and temperatures, which are significant for assessing the machinability of the workpiece material and predicting tool wear. In this paper, the material flow in orthogonal cutting, including primary and secondary shear zones, is represented by a viscous/viscoplastic model that includes the temperature-sensitive Johnson-Cook flow stress model. A stabilized staggered finite element procedure is developed to handle incompressible Navier-Stokes material flow in combination with convection-dominated hardening and thermomechanical interaction. To handle material flow at tool-workpiece contact, a mixed method is used to reduce spurious oscillations in contact stresses along with simplified heat transfer in the tool-workpiece interface. A novel feature is that the velocity field is resolved as a subscale field to the velocity field of the distributed primary zone deformation model. It appears that the finite element solution to the subscale material flow model is significantly more cost-effective in contrast to directly addressing the velocity field and compared to the chip-forming simulations (DEFORM 2D). The cutting forces, temperature, and stress-strain state of the material in the critical deformation regions can be accurately estimated using the subscale model. The results obtained show that the trend of the estimated forces and temperatures is consistent with our experimental measurements, the DEFORM 2D simulations, and the experimental data from the literature.

Rate-independent model of ferroelectric materials: finite element and finite difference solution

Rate-independent model of ferroelectric materials: finite element and finite difference solution

Effect of spatial variability of soil properties on slopes stability under rapid water drawdown conditions

Stability analysis of slopes is a fundamental problem of Soil Mechanics. Its main objective consists of evaluating the potential failure of the slope structure under a prescribed loading mode. A major component of the latter refers to the seepage forces induced by pore-pressure gradient, which is known to be responsible of destabilizing effects. This contribution employs a kinematic approach of limit analysis theory to obtain upper-bounds solutions to the stability problem of saturated slopes submitted to rapid water level drawdown. Random fields are used to model the uncertainty surrounding the spatial distribution of soil cohesion, friction angle, and permeability. In the context of effective stress governing the strength capacities of the soil material, it is shown that the seepage forces related to the water flow regime can be considered as external volumetric loads in the assessment of stability. The hydraulic problem governing the water filtration velocity is evaluated by resorting to an analytical variational approach, whose results are validated by comparisons with finite element solutions. The impact of hydraulic-related parameters on stability is first investigated for slopes within a deterministic framework. Subsequently, random fields are considered to take into account the variability related to the spatial distribution of the material properties, which are discretized using the Karhunen-Loève expansion with numerically computed eigenfunctions. The failure probability of slopes is assessed through Monte Carlo simulations. Several analyses have been performed to discuss the influence of the spatial variability of relevant problem parameters.

Airfoil lift finite element computations using H1 (Ω) and H(div,Ω) approximation spaces

Computing the lift generated by an airfoil is crucial in aircraft design, especially for aircraft performance prediction, aircraft stability/control, and optimization of airfoil shape for improving efficiency. The problem of incompressible flow around an airfoil is studied herein as a weakly irrotational flow, resulting a div-curl problem in a 2D double-connected domain. The original div-curl problem is ill-posed because it is defined over a multiply-connected domain. Traditionally, a cut in the original domain and additional constrain for the velocity potential and velocity potential gradient at the cut are required for the numerical solution, which applies a rotational correction method based on Helmholtz decomposition of vector fields. The lift coefficient can be finally computed from the numerical solution of the problem. In the present work, the lift of an airfoil is computed from finite element solutions using different approximation spaces: The conventional H1 (Ω) potential-field space and a special class of H(div,Ω) velocity-field space, i.e., the divergence-free space. Accurate analysis using both approximations is performed with h-refinement strategies. The lift coefficient computed from the analysis is compared against available reference solutions. The computational performance and accuracy of the lift computation using the two different approaches are discussed. Finally, the difference in the velocity solution obtained using H1 (Ω) and H(div,Ω) spaces is presented, which can be interpreted as a new posteriori error estimator for the problem.

Novel Volume Integral Equation Approach for Low-Frequency E-Field Dosimetry of Transcranial Magnetic Stimulation.

A new volume integral equation (VIE) approach is introduced to study transcranial magnetic stimulation (TMS) and high-contrast media at low frequencies. This new integral equation offers a simple solution to the high-contrast breakdown observed in low-frequency electric field (E-field) dosimetry of conductive media. Specifically, we employ appropriate approximations that are valid for low frequencies and stabilize the VIE by introducing a basis expansion set that removes solutions associated with high eigenvalues in the equation. The new equation is devoid of high-contrast breakdown and does not require the use of auxiliary surface variables or projectors, providing a straightforward practical solution for the VIE analysis of TMS. Our results indicate that the novel VIE formulation matches boundary element, finite element, and analytical solutions. This new VIE represents a first step towards including anisotropy in integral equation E-field dosimetry for brain stimulation.

Development of an analytical solution to diffusion equation for multidimensional solids of arbitrary shape

Mass transfer processes between solids and fluids are ubiquitous in the food engineering field and using simple analytical models for their simulation or estimation of mass diffusivities remains widespread. Unfortunately, these solutions are available only for a limited number of simple solid geometries which might not be applicable to all food shapes; thus, this study introduces a simple method to obtain the volume average concentration in multidimensional solids of arbitrary shape (SAS). The proposed method approximates the (analytical or numerical) solution of the SAS from the analytical solution of a properly sized box by minimizing a weighted similitude index between the original and box shapes. Besides, the method was generalized to consider the isotropic shrinkage of foods (that is, size reduction while maintaining the aspect ratio). The applicability of the equivalent box approach was exemplified by solving inverse problems for the estimation of caffeine diffusivity during aqueous extraction of green coffee beans and water diffusivity during lentil drying using data available from literature. The results were compared with those obtained by the finite element solution of the 3D mass transfer model using the real product shape. Mass diffusivities for caffeine in green coffee and for water in lentils estimated with the equivalent box approach were not statistically different (p>0.05) to those estimated by numerically solving the 3D mass transfer model under the same assumptions. This method represents a simple and reliable way to model mass transfer in complex-shaped foods.

Finite element solution to the Poissonian irradiance transport equation applying structured patterns in SLM for wavefront sensing

We present a new technique, to our knowledge, to obtain the wavefront. We propose to modify the irradiance transport equation (ITE) by using fringe patterns of spatial light modulators (SLMs) and super-Gaussian Ronchi rulings (SG-RRs) to create the Poissonian irradiance transport equation (PITE) to solve the wavefront using the finite element method (FEM). We use a liquid crystal spatial light modulator (LCSLM) to build periodic patterns, which permits simplification of the irradiance transport equation (ITE) into a look-a-like Poisson’s equation under experimental conditions. First, we model different flat/parallel patterns based on super-Gaussian (SG) profiles with different frequencies optimizing the results when substituting the conventional Ronchi rulings in arrays for wavefront sensing. We then analyze the noise reduction in the experimental irradiance captures with the use of SG periodic profiles in LCSLM to induce periodicity in different irradiance distributions. We also analyze the difference between irradiance captures (according to the ITE) to obtain the PITE, which we subsequently solve as a Poisson’s equation applying the finite element method (FEM), with triangular symmetry between the mesh nodes. We do this for two meshes, having a different number of nodes, and obtaining for each the wavefront surface [W(ρ,θ)]. We characterize the aberrations in W(ρ,θ) by means of a multilinear fit of the Zernike polynomials (Zi,j) with degree M=50 to optimize the comparison between the main aberration coefficients in evaluation curves and to minimize the dispersion with comparable results obtained from software tools devoted to the analysis of interferograms. These software tools are APEX, FringeXP, and OpenFringe. Finally, we discuss our results.

Finite element solutions of Jeffery fluid considering Xue-and Yamada-Ota model on circular cylinder with a thermal jump

The current investigation is based on FEM (finite element approach) simulations of Jeffery fluid in heat transfer analysis on a circular cylinder subjected to convective boundary conditions. The mathematical model of the X- and YO-model is formulated to investigate the impacts of thermal efficiency. Kerosine oil is assumed to be a base fluid along with nanofluid silicon dioxide and titanium dioxide. Heat transfer occurs through the effects of heat source and quadratic thermal radiation. The objective of the current model is to determine the comparative thermal performance between Xue- and Yamada-Ota models utilized in solar energy, biomedicines electronic processes, etc. It is demonstrated that such a developing model has not yet been studied using finite element methodology. Yamada-Ota hybrid nano-fluid model has maximum thermal production as compared to thermal production for Xue-hybrid nano-fluid. Skin friction coefficients increase when mixed convection and radiation numbers are enhanced.

Two-grid finite element methods for space-fractional nonlinear Schrödinger equations

A two-grid finite element method is developed for solving space-fractional nonlinear Schrödinger equations. The finite element solution in L∞-norm is proved bounded without any time-step size conditions (dependent on spatial-step size). Then, the optimal order error estimations of the two-grid solution in the Lp-norm are proved without any time-step size conditions. Finally, the theoretical results are verified by numerical experiments.

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh–Stokes equations

In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh–Stokes equations. The proposed two-grid FEM uses the L2-1σ scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The L2-norm and H1-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy H=h12 and H=hr2r+2 respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-1σ scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.

A reduced-dimension method of Crank-Nicolson finite element solution coefficient vectors for the unsteady Burgers equation

This paper primarily focuses on the dimensionality reduction of finite element (FE) solution coefficient vectors for the unsteady Burgers equation, solved using the Crank-Nicolson FE (CNFE) method. The proper orthogonal decomposition (POD) basis is constructed from the snapshot matrix, which is formed using the first L solutions, where L is significantly smaller than the total number of time steps N of the CNFE method. By reconstructing the matrix form of the CNFE method, a reduced-dimension Crank-Nicolson finite element (RDCNFE) method is proposed and stability analysis and error estimates are discussed. Numerical tests are implemented to verify the theoretical results and demonstrate the high efficiency of the RDCNFE method.

Efficient layerwise multiphysics spectral element model for delaminated composite strips with PWAS transducers

Efficient structural element-based models are essential for fast simulations of wave propagation in composite structures for model-based and physics-informed data-driven structural health monitoring. This article introduces the first efficient multiphysics time-domain spectral structural element for wave propagation analysis of beam and panel-type composite structures with piezoelectric transducers (patch or full layers) containing delaminations. A general framework is presented to model multiple delaminations and transducer patches located arbitrarily. The intact host and patch transducer-bonded laminates and sub-laminates between delaminations are modelled separately using an electromechanically coupled efficient layerwise zigzag theory (ZIGT) for kinematics and a piecewise quadratic variation for the electric potential across piezoelectric layers. The high-order spectral element (SE) features a virtual electric node to model equipotential surfaces of piezoelectric transducers apart from the usual physical nodes having mechanical and internal electric degrees of freedom. A hybrid point-least squares continuity approach is employed to maintain continuity at the intersections of delaminated sub-laminates or the patch-bonded laminate with the host laminate. The model’s performance in capturing electroelastic waves’ interaction with delamination is examined with reference to the conventional finite element (FE) solution based on the ZIGT, continuum-based FE solutions, and the SE solution based on a non-layerwise version of the laminate theory. Finally, the model is used to examine the impact of interfacial location and size of delaminations on wave propagation behaviour.

Two-grid reduced-dimension extrapolated method of Crank-Nicolson finite element solution coefficient vectors for nonlinear parabolic equation

Two-grid reduced-dimension extrapolated method of Crank-Nicolson finite element solution coefficient vectors for nonlinear parabolic equation

A novel post-processed finite element method and its convergence for partial differential equations

In this article, by combining high-order interpolation on coarse meshes and low-order finite element solutions on fine meshes, we propose a novel approach to improve the accuracy of the finite element method. The new method is in general suitable for most partial differential equations. For simplicity, we use the second-order elliptic problem as an example to show how the novel approach improves the accuracy of the finite element method. Numerical tests are also conducted to validate the main theoretical results.

Peridynamics model of torsion-warping: Application to lattice beam structures

This paper presents a finite deformation beam model based on Simo-Reissner theory in peridynamics (PD) framework to deal with torsion induced warping deformation. Seven degrees of freedom, viz. three translational, three rotational, and one warping amplitude are considered at each material point. The governing equations of the beam are obtained by employing global balance of linear and angular momenta in conjunction with Simo's assumption on the deformation field. The relation between PD resultant force, moment, bi-moment, and bi-shear states with their classical counterparts is established using the constitutive correspondence method. Numerical implementation strategy is furnished for both quasi-static and dynamic cases. The solution for quasi-static load is obtained through the Newton-Raphson method. The proposed model is validated against finite element solutions considering cantilever beam and lattice structures. Quasi-static deformation responses of 3 × 3 × 3 octet and single unit compression-torsion lattice structures are presented further to demonstrate the effectiveness of proposed beam model. A new bond breaking criterion is proposed based on critical stretch, critical relative rotation, and critical relative warping amplitude and failure of the compression-torsion lattice structures under compressive load is simulated. The Newmark-beta method is utilized to solve the governing equations for dynamic loading. Numerical simulations include dynamic analysis of octet and compression-torsion lattice structures.

A plastic correction algorithm for full-field elasto-plastic finite element simulations: critical assessment of predictive capabilities and improvement by machine learning

AbstractThis paper introduces a new local plastic correction algorithm that is aimed at accelerating elasto-plastic finite element (FE) simulations for structural problems exhibiting localised plasticity (around e.g. notches, geometrical defects). The proposed method belongs to the category of generalised multi-axial Neuber-type methods, which process the results of an elastic prediction point-wise in order to calculate an approximation of the full elasto-plastic solution. The proposed algorithm relies on a rule of local proportionality, which, in the context of J2 plasticity, allows us to express the plastic correction problem in terms of the amplitude of the full mechanical tensors only. This lightweight correction problem can be solved for numerically using a fully implicit time integrator that shares similarities with the radial return algorithm. The numerical capabilities of the proposed algorithm are demonstrated for a notched structure and a specimen containing a distribution of spherical pores, subjected to monotonic and cyclic loading. As a second point of innovation, we show that the proposed local plastic correction algorithm can be further accelerated by employing a simple meta-modelling strategy, with virtually no added errors. At last, we develop and investigate the merits of a deep-learning-based corrective layer designed to reduce the approximation error of the plastic corrector. A convolutional architecture is used to analyse the neighbourhoods of material points and outputs a scalar correction to the point-wise Neuber-type predictions. This optional brick of the proposed plastic correction methodology relies on the availability of a set of full elasto-plastic finite element solutions to be used as a training data-set.

Anisotropic error estimator for the Stokes–Biot system

This paper presents an a posteriori error analysis for the problem defining the interaction between a free fluid and poroelastic structure approximated by finite element methods on anisotropic meshes in Rd, d=2 or 3. Korn’s inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to nonconforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for conforming and nonconforming cases. With the obtained finite element solutions, local error indicators and a global estimator are generated, demonstrating reliability and efficiency. The efficiency is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the reliability, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media.

A novel dimensionality reduction iterative method for the unknown coefficient vectors in TGFECN solutions of unsaturated soil water flow problem

The main purpose of this paper is to reduce the dimensionality of unknown coefficient vectors in finite element (FE) solutions of two-grid FE Crank-Nicolson (CN) (TGFECN) format for the unsaturated soil water flow (USWF) problem with two strong nonlinear terms by using proper orthogonal decomposition (POD). For this purpose, our first step involves designing a time semi-discrete CN (TSDCN) scheme for the USWF problem and demonstrate the existence, boundedness, and error estimations of TSDCN solutions. Thereafter, we discretize the TSDCN scheme using the two-grid FE method to create a new TGFECN format and prove the existence, boundedness, and error estimations of TGFECN solutions. The primary focus should be on reducing the dimension of unknown coefficient vectors of TGFECN solutions through the utilization of the POD technique in creating a novel format, referred to as dimension reduction iterative TGFECN (DRITGFECN) format, while establishing the existence, boundedness, and error estimations for DRITGFECN solutions. Lastly, we use two sets of numerical tests to exhibit the advantage of the DRITGFECN format. Due to the presence of two highly nonlinear terms in the unsaturated soil flow problem, the development and analysis of DRITGFECN format pose greater challenges and necessitates more advanced technical skills compared to previous studies. However, the significance and broad applications of this research make it a valuable subject for investigation.

A Gauss kernel non-local stress-driven plate theory

This study presents a novel stress-driven formulation, incorporating Kirchhoff’s plate assumptions and accounting for the non-local size-dependent feature in both plate geometric axes, which is practical, especially near the origin of the axis in the polar coordinates. The presented formulation does not impose any additional constraints on the elastic radial curvatures, which are known as constitutive boundary conditions. Therefore, taking into account the non-local behavior in both radial and circumferential directions does not lead to an overconstrained problem. The study focuses on the analysis of a circular plate with an outer radius (R), thickness (h), and the length scale parameter (L). The governing differential equations are derived in Cartesian coordinates and then transformed to the polar coordinates to analyze circular plates. When R tends to infinity, the governing equation is solved analytically. Using the Galerkin technique, a finite element solution is presented for the analysis of a bounded circular plate with a clamped boundary condition at the outer radius R. The study discusses the size effects and the gradient properties of functionally graded plates in stress-driven non-local theory. It reveals that while increasing the non-local parameter results in a reduction of transverse displacement, the moments remain unaffected.

New Analytical Solutions for Constant Rate Pumping in Two‐Zone Double‐Porosity Confined Aquifer: A New Source Term Reflecting Effects of Well Skin and Wellbore Storage

AbstractThis study develops two new analytical models for constant rate pumping at a partially penetrating well in a double‐porosity confined aquifer, considering skin and formation zones. One model, referred to as the two‐zone model, incorporates a flow equation to depict the flow in the skin around the well. The other model, named the source‐term model, introduces a novel source term at the outer rim of the skin to reflect the effects of both the skin and wellbore storage. The analytical solutions for both models are derived by the Laplace transform and finite Fourier cosine transform. Additionally, a finite element solution for the source‐term model is presented. Results suggest the source‐term model is suitable to most wells when the width of the skin is less than 1 m and the radius of influence exceeds the outer rim of the skin. Temporal drawdown distribution for a negative skin exhibits a triple‐humped shape with two flat stages, while that for a positive skin shows monotonous increase. The source‐term model enables orthogonal 5 × 5 nodes for finite element approximation to discretize a well and its adjacent skin. The finite element solution aligns with early drawdown data measured at an observation well under the effects observed in two field constant rate pumping tests. In conclusion, this study introduces a novel approach to modeling two‐zone flow, which may find practical utility in field applications.