Finite Element Method Mesh Research Articles

Adaptive Finite Element Model for Simulating Crack Growth in the Presence of Holes.

This study presents a developed finite element code written by Visual Fortran to computationally model fatigue crack growth (FCG) in arbitrary 2D structures with constant amplitude loading, using the linear elastic fracture mechanics (LEFM) concept. Accordingly, optimizing an FCG analysis, it is necessary to describe all the characteristics of the 2D model of the cracked component, including loads, support conditions, and material characteristics. The advancing front method has been used to generate the finite element mesh. The equivalent stress intensity factor was used as the onset criteria of crack propagation, since it is the main significant parameter that must be precisely predicted. As such, a criterion premised on direction (maximum circumferential stress theory) was implemented. After pre-processing, the analysis continues with incremental analysis of the crack growth, which is discretized into short straight segments. The adaptive mesh finite element method was used to perform the stress analysis for each increment. The displacement extrapolation technique was employed at each crack extension increment to compute the SIFs, which are then assessed by the maximum circumferential stress theory to determine the direction of the crack growth and predict the fatigue life as a function of crack length using a modified form of Paris’ law. The application examples demonstrate the developed program’s capability and performance.

A thermal fluid mechanical model of stress evolution for wire feeding-based laser additive manufacturing

Wire feeding-based laser additive manufacturing (WFLAM), highlighting high flexibility and efficiency, is widely used in large-scale structures. However, the heat input accumulation and large-scale effect lead to severe stress and deformation in the large-scale structures. This paper presented a novel coupled thermal fluid mechanical model of stress evolution considering the molten pool fluid flow influence by combining the computational fluid dynamics (CFD) model and the finite element method (FEM) model. In the CFD model, the fluid flow and heat transfer are considered; in the FEM model, the thermo-elastoplastic theory is used based on the non-linear material parameters. An accurate mapping algorithm was proposed to transfer the volume of fluid (VOF) and the temperature value from the CFD data to the FEM mesh. This model could obtain a more accurate temperature field than the thermal-mechanical model with the regular heat source. Besides, the model could consider the influence of the shape on the stress accurately. The simulation results show the tensile stress on the root can be increased from 370 MPa to more than 800 MPa, with the influence of the sharp angle defects. At the sidewall, the influence of the surface angle on the residual Mises stress is around 1100 MPa/rad.

Error-Controlled Static Layered-Medium Green’s Function Computation via hp-Adaptive Spectral Differential Equation Approximation Method

A numerically robust and computationally efficient approach for evaluating a planar layered substrate's static Green's function is developed based on the adaptive form of the spectral differential equation approximation method. The method uses a pth-order finite element method (FEM) solution of the 1-D ordinary differential equation governing the spectrum of the layered-media Green's function with spatial h-adaptive meshing. The resulting pole-residue form of the Green's function spectrum enables analytic evaluation of the pertinent Sommerfeld integrals providing O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> ) error control of the spatial layered-medium Green's function in near, intermediate, and far zones. The detailed error analysis is presented enabling automation of the 1-D FEM mesh refinement, which guarantees a prescribed accuracy of the solution depending on the distance between the source and observation locations. The method is well suited for computing Green's function databases used by method of moments capacitance and inductance extractors.

A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier–Stokes problems

We focus on steady and unsteady Navier–Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1–3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.

A probabilistic finite element method based on random meshes: A posteriori error estimators and Bayesian inverse problems

We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a probability measure on classical FEMs to quantify the uncertainty due to numerical errors either in the context of a-posteriori error quantification or for FE based Bayesian inverse problems. The new approach involves only a perturbation of the mesh and an interpolation that are very simple to implement We present a posteriori error estimators and a rigorous a posteriori error analysis based uniquely on probabilistic information for standard piecewise linear FEM. A series of numerical experiments illustrates the potential of the RM-FEM for error estimation and validates our analysis. We furthermore demonstrate how employing the RM-FEM enhances the quality of the solution of Bayesian inverse problems, thus allowing a better quantification of numerical errors in pipelines of computations.

An enriched finite element method with interpolation cover functions for acoustic analysis in high frequencies

At present, the conventional finite element method (FEM) still has some defects for solving acoustic problems governed by the Helmholtz equation. Owing to the numerical dispersion error, the standard FEM always fails to provide sufficiently satisfactory results in high frequency range. To cure this drawback, an enriched finite element method (E-FEM) with interpolation cover functions is presented to deal with 2D and 3D acoustic problems in this work. In the present E-FEM, the nodal interpolation functions from the standard FEM are enriched by the additional interpolation cover functions over patches of elements to improve the performance (such as convergence rate and computation accuracy) of the standard FEM and the original mesh (such as the conventional triangular and tetrahedral mesh) in the standard FEM can still be used. In addition, the high gradient components of the considered field variable can be captured and the inter-element gradient jumps can be smoothed out by the used interpolation cover functions, so the present E-FEM is able to provide a smoother and more accurate acoustic field for acoustic analyses. Several numerical examples demonstrate that higher computational efficiency and accuracy can be obtained for analyzing acoustic problems compared to the conventional FEM. Moreover, the E-FEM can effectively suppress the dispersion error at high frequencies, so it can make an effective prediction for practical engineering problems.

Structural analysis of composite tubes using a meshless analytical dimensional reduction method

AbstractA polynomial‐based model in conjunction with dimensional reduction method are presented to perform cross‐sectional analysis and to determine strain distribution in composite tubes under bending loading. For beam structures with tubular cross‐section, the Variational Asymptotic Method (VAM) has been employed to decompose a three‐dimensional (3D) elasticity problem into a two‐dimensional cross‐sectional analysis and a one‐dimensional analysis along the length. This greatly reduces computational time as compared to 3D Finite Element Method (FEM). There also exists publically available Variational Asymptotic Beam Sectional Analysis (VABS), a FEM cross‐sectional analysis tool based on VAM. For VABS, FEM mesh for the beam section needs to be generated. For the case of composite tubes with many layers, the mesh generation consumes efforts and is unnecessary. We introduce a new meshless dimensional reduction method for the analysis of composite tubes. This method utilizes Pascal polynomials in polar coordinates to model the warping functions. Using this, one can obtain stiffness constants and 3D strains of the composite tube. This method is straightforward, meshless, with similar computation time as VABS, and is much more efficient than conventional 3D FEM. The accuracy of the proposed method is examined by comparing the obtained results with 3D FE (ANSYS), VABS, literature, and experiment.

New 3D Diffusion Code Based on The Nodal Polynomial Expansion

Nodal Expansion Method (NEM) is widely employed in the neutronic design of nuclear reactors core. The main reason is its higher computational performance and efficiency when compared with the conventional fine mesh difference method or finite element method. The NEM diffusion calculation uses coarse spatial meshes and the size can lie in assembly scale. This is the key for the computational efficiency and high accuracy calculations. The NEM consists mainly of basis polynomial function expansion for each nodal direction. The Nodal Expansion Method (NEM), as proposed by Finnemann in 1977, has been used to solve the multigroup neutron diffusion equations in three-dimensional (3D) rectangular geometry. The weighted residual technique has been applied to determine the higher order coupling coefficients. Resulting from the combination of nodal and finite element methods, NEM provides rigorously accurate equations obtained by integrating the neutron balance equation. The two group coarse mesh 3D IAEA benchmark has been simulated by NEM3D-1A using different nodes sizes. The VENTURE

Simulation of Quasi-Static Crack Propagation by Adaptive Finite Element Method

The finite element method (FEM) is a widely used technique in research, including but not restricted to the growth of cracks in engineering applications. However, failure to use fine meshes poses problems in modeling the singular stress field around the crack tip in the singular element region. This work aims at using the original source code program by Visual FORTRAN language to predict the crack propagation and fatigue lifetime using the adaptive dens mesh finite element method. This developed program involves the adaptive mesh generator according to the advancing front method as well as both the pre-processing and post-processing for the crack growth simulation under linear elastic fracture mechanics theory. The stress state at a crack tip is characterized by the stress intensity factor associated with the rate of crack growth. The quarter-point singular elements are constructed around the crack tip to accurately represent the singularity of this region. Under linear elastic fracture mechanics (LEFM) with an assumption in various configurations, the Paris law model was employed to evaluate mixed-mode fatigue life for two specimens under constant amplitude loading. The framework includes a progressive analysis of the stress intensity factors (SIFs), the direction of crack growth, and the estimation of fatigue life. The results of the analysis are consistent with other experimental and numerical studies in the literature for the prediction of the fatigue crack growth trajectories as well as the calculation of stress intensity factors.

Permanent Charge Effects on Ionic Flow: A Numerical Study of Flux Ratios and Their Bifurcation

Ionic flow carries electrical signals for cells to communicate with each other. The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flow properties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planck model for small and relatively large permanent charges. The analytical studies have led to the introduction of flux ratios that reflect permanent charge effects and have a universal property. The studies also show that the flux ratios have different behaviors for small and large permanent charges. However, the existing analytical techniques can reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios to bridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. In particular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numerical techniques. An adaptive moving mesh finite element method is used in the numerical studies.

Error analysis of the unstructured mesh finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential

In this paper, error analysis of the unstructured mesh Galerkin finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential defined on a finite domain is studied. The finite difference method is used to discretize the Caputo time fractional derivative, while the finite element method using unstructured mesh is used to deal with the Riesz fractional operators in space. Both the stability and convergence analysis of the numerical scheme are constructed. Numerical example is conducted to testify the validity of the proposed method. The conservation of the space fractional Schrödinger equation and the non-conservation of the time fractional Schrödinger equation in quantum mechanical system are achieved. This paper proposes an efficient numerical method as well as its theoretical analysis for the two-dimensional time-space fractional Schrödinger equation with time-independent potentials.

Pressure Wave Caused by Trinitrotoluene (TNT) Underwater Explosion—Short Review

The development of computational techniques and computer hardware has an impact the analysis of short-term (fast-changing) processes, such as the impact of a non-contact underwater explosion pressure waves. A theory of underwater explosions, gas bubble formation and pressure waves are presented. The course of the pressure wave in time, and its propagation in the acoustic medium are presented. The study presents empirical descriptions of non-contact pressure explosion waves. We propose to use them in simulations of ship hull strength and other objects immersed in liquids that are exposed to the effects of non-contact trinitrotoluene (TNT)-charge explosions. Pressure distributions and their time courses given by authors such as R.H. Cole, J.S. Nawagin, W. Stiepanow, T.E. Farley and H.G. Snay, T.L. Geers and K.S. Hunter are compared. A method of pressure wave modeling using acoustic media implemented in Computer Aided Engineering (CAE) programs is presented. The results of the values and the time course of the pressure acting on the underwater object are given. The influence of FEM (Finite Element Method) mesh density on the obtained results is examined and presented. The aim of the article is to expand our knowledge of underwater explosions, compare mathematical descriptions of the pressure waves developed by different authors and show the differences between them. In addition, we present the distinction between contact and non-contact explosions and analyze how changes in the mesh density of acoustic elements affects the reflection of the incident wave caused by an underwater explosion.

Regularization of continuum damage mechanics models for 3-D brittle materials using implicit gradient enhancement

Abstract A well-known problem which arises in local damage models is that they suffer from mesh dependence and strain localization. In this study, we employ the non-local implicit gradient damage formulation for mesh sensitivity analysis of brittle materials. The basic concept of the non-local implicit gradient model is that the strain at a given point depends not only on the strain at that point but also on the nearby strain field. The local equivalent strain is replaced with a non-local equivalent strain, which is constructed by solving a system of partial differential equations (PDEs). We solve this system of PDEs using finite element method (FEM) and split-operator method which has been applied to 3-D microplane damage models. We extend this approach to continuum damage models and apply it to brittle material for three different loading experiments. In these examples, we demonstrate that the implementation of the implicit gradient damage model using the split-operator method can successfully be applied to complex, 3-D geometries requiring large-scale unstructured FEM meshes, and eliminate mesh dependence. In not all cases the simulations were able to obtain the desired fracture widths as this would require meshes at even smaller resolutions which is beyond our current computing capacity.

An adaptive moving finite element method for steady low Mach number compressible combustion problems

SummaryThis work surveys an r‐adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the computation of such a problem is often extremely time‐consuming. Importantly, to capture the significant characteristics of the flame structure when using detailed chemistry, a much more stringent requirement on the spatial resolution of the interior layers of some intermediate species is necessary. Here, we propose a moving mesh method in which the mesh is obtained from the solution of so‐called moving mesh partial differential equations. Such equations result from the variational formulation of a minimization problem for a given target functional that characterizes the inherent difficulty in the numerical approximation of the underlying physical equations. Adaptive mesh movement has emerged as an area of intense research in mesh adaptation in the last decade. With this approach, points are only allowed to be shifted in space leaving the topology of the grid unchanged. In contrast to methods with local refinement, data structure hence is unchanged and load balancing is not an issue as grid points remain on the processor where they are. We will demonstrate the high potential of moving mesh methods for effectively optimizing the distribution of grid points to reach the required resolution for chemically reacting flows with extremely thin boundary layers.

Development of the triangle-based nodal algorithm for reconstructing pin power distributions

The fuel pin power is an essential parameter for increasing the safety and reliability features of the reactor. Although the high-order triangle-based polynomial expansion nodal (TPEN) algorithm had been suggested for incredible accuracy and computational speed in hexagonal core analyses, the pin power reconstruction of this algorithm is not developed and assessed. This paper presents the characteristics and performances of the TPEN algorithm for pin power reconstruction. The converged nodal information obtained from sweep between coarse-mesh finite difference (CMFD) and TPEN methods is used to reconstruct the neutron flux distribution in a homogeneous fuel assembly. The modulation technique is used to obtain the heterogeneous distribution of power density. In this technique, the power density homogeneous distribution, calculated with the reconstructed neutron flux is multiplied by a form function. These functions are generated by DRAGON5. The results obtained by this algorithm are verified for various core configurations of a VVER-1000 reactor. The pin power factors show good agreement with the reference solution obtained by heterogeneous fine mesh finite element method. The largest and average relative errors found were of the order of 5% and 0.5% for TPEN, in a peripheral cell of a fuel element with the faces towards the region of the baffle/reflector. We also compared the results with those obtained from the nodal expansion method (NEM). The maximum and mean relative error of 10% and 1% are found by the NEM method.

Coupling of peridynamics with finite elements for brittle crack propagation problems

In this work, a new coupling model of finite element method (FEM) and ordinary state-based peridynamics (OSBPD) is proposed to accurately predict brittle fracture under quasi-static and dynamic loading. PD grids are directly coupled with FEM meshes without an overlapped zone. The transferred forces between PD grids and FEM meshes are reciprocal and balanced inherently, which avoids the complexity of the transitional zone-based coupling models and the non-equilibrium phenomenon of the structural force reported in the sharing nodes-based coupling model. More importantly, the proposed coupling model is free of significant influence of ghost forces and significant wave reflection at the coupling interface. Also, the surface effect of PD can be approximately removed. The accuracy of the proposed coupling model is verified through comparing with the FEM results and pure OSBPD results.

New Algorithm for Three-Dimensional Borehole-to-Surface Apparent Resistivity Imaging Based on Unstructured Mesh Finite-Element Method

In this paper, a finite-element method with unstructured mesh is used to numerically simulate the line source model of three-dimensional borehole-to-surface electrical potential method. A new total field potential is synthesized using the calculated anomalous field potential and the analytic solution of the background field. In solving for the original total field potential and background potential, this method can guarantee a small near-field error in the new total field potential without the need to densely mesh the region with the source. By comparing the synthesized total field and analytic solution, the high accuracy of the numerical solution is verified. The results of model calculation show that the detection effectiveness of the borehole-to-surface electrical method depends on the length of the line source. The detection effectiveness increases substantially with increasing length of the line source within a certain range. However, when the line source is excessively long, the increase in detection effectiveness for anomalies is insignificant. For a low-resistance anomaly of a given volume, the borehole-to-surface electrical method is more sensitive to changes in the lateral area than to changes in the thickness. When the gob contains only a small amount of water (e.g., Model IV containing 5% of water by volume), it will appear as a low-resistance anomaly. In addition, the detection of a water-logged gob by the borehole-to-surface electrical method depends on the depth and size of the gob. The borehole-to-surface method can accurately delineate the profile boundary of an anomaly, even though the side closer to the source is susceptible to the effect of source current. These results are significant for guiding the field exploration of the borehole-to-surface electrical method, improving result interpretation, and assisting the application.

Hybrid FEM and FDM Approach for Monolithic Catalytic Converters with Pore Diffusion

A transient, three-dimensional model is developed for monolith catalytic converters with washcoat pore diffusion. Due to the unique configuration of monolith converters, the flow within the channel can be considered fully developed laminar flow, and one-dimensional transport equations can be formulated for the flow and species. On the other hand, a three-dimensional equation is necessary for the monolith solid temperature to capture the effects of significant external heat loss or flow maldistribution. The proposed model employs a three-dimensional finite-element method (FEM) for the solid temperature equation and a one-dimensional finite-difference method (FDM) for the flow and species equations. A recently developed asymptotic approach (Bissett, Emiss. Control Sci. Technol. 1(1), 3–16, 2015, Bissett, Emiss. Control Sci. Technol. 5(1), 45–54, 2019) is utilized to solve the washcoat pore diffusion and surface coverages equations. The three-dimensional FEM meshes are chosen so the nodes also serve for the one-dimensional transport equations. All equations are solved fully coupled, without lagging or reliance upon external tools. The developed solver can efficiently simulate general cross-section geometries (circle, oval, rectangle, etc.) for a catalytic monolith under nonuniform inlet distribution and detailed washcoat pore diffusion. The solver is verified by comparing the simulation results to the one-dimensional simulation results for a simple geometry under uniform inlet conditions. The model can be applied to flow through general monolith catalytic reactors with single or dual washcoat layers and a wide range of kinetics.

Single-fiber pull-out analysis comparing the intensities of singular stress fields (ISSFs) at fiber end/entry points

This paper deals with a partially-embedded single-fiber under pull-out force in comparison with a single fiber embedded in matrix focusing on two distinct singular stress fields. Then, the intensities of the singular stress fields (ISSFs) are compared at the fiber end named Point A and the fiber/surface intersection named Point E. The results show that if the embedded length lin is shorter, interface debonding may occur at Point A. Instead, if lin is longer, the interface debonding may occur at Point E. To analyze the ISSFs accurately, a mesh-independent technique coupled with the finite element method (FEM) is indicated by applying the same FEM mesh pattern to the pull-out model and the reference model. As the reference solution, a single fiber embedded in matrix is also calculated under arbitrary material combinations by using the body force method (BFM). Stress distributions along the fiber/matrix interfaces are also calculated for carbon and glass fibers.

Projection-based reduced order models for a cut finite element method in parametrized domains

This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.