High-order Finite Element Research Articles

High-order, unconditionally maximum-principle preserving finite element method for the Allen–Cahn equation

In this paper, based on the mass-lumping finite element space discretization, we incorporate the integrating factor Runge–Kutta method and stabilization technique to develop a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen–Cahn equation and its conservative forms. The proposed methods are linear, without requiring any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Several numerical experiments verify the high-order temporal accuracy of the proposed schemes, as well demonstrate the ability to preserve the maximum principle, mass conservation, and energy stability over long periods. Moreover, by the aid of numerical simulation, we show that the proposed schemes also have good performances in terms of structure-preserving with high order finite element method.

On effects of concentrated loads on perforated sensitive shells of revolution

Sensitive shells are a class of constructions with specific and perhaps unintuitive responses to different loading scenarios. There are new emerging applications for capsules with open cavities, and as the designs are optimised to maximise the payload, sensitivity has to be taken into account. Concentrated loads induce singularities that propagate along the characteristics of the surfaces. Perforation patterns do not have a strong effect on internal layers. Under a symmetric concentrated loads for parabolic and hyperbolic shells there exists a critical thickness at which local features of solution begin to dominate, whereas for elliptic cases there is always a dominant global response. In non-uniform curvature cases the elliptic part will dictate the solution even if the load is not acting on it. The extensive set of simulations has been computed using high-order finite element solvers including adaptivity.

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.

Modeling the performance of a family of phononic pseudo-crystal interposers

Recent technical efforts at Sandia National Laboratories have uncovered a need for high-frequency isolation in solid media across a range of ultrasonic frequencies; these needs include the requirement to suppress crosstalk between piezo-mechanical data and power channels situated on the same substrate. Broadband isolation larger than is possible with comparable phononic crystal designs is provided using a recently-developed phononic pseudo-crystal isolator concept. Insertion loss results from 2-D finite element simulations in COMSOL and 3-D simulations in Sierra-SD, Sandia’s HPC compatible structural dynamics simulation platform, are used to characterize the performance of characteristic examples of this family of phononic isolators. Sierra-SD uses massively parallel computing and high-order polynomial finite elements (P-elements) to mitigate the computational cost required to simulate coupled electrical-mechanical 3-D ultrasonic wave propagation. [SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525.]

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method

Optimal sensor placement and estimator-based temperature control for a deep drawing process

Deep drawing is one of the most important forming processes for the forming of flat sheet blanks, where the formation of wrinkles and the appearance of cracks can be a problem, especially in areas of high geometric complexity. The local increase of the temperature in these critical areas can help to improve the formability of the material and thus reduce defects. The present paper aims at a targeted temperature control of the die of a deep-drawing mold. For this sensors are placed systematically to develop an estimator for the spatial–temporal temperature evolution to subsequently realize tracking control using the embedded actuation devices. A continuum representation of the temperature distribution in the die is derived and transferred to a high order finite element (FE) approximation to take the complex-shaped geometry of the tool into account. Parameter identification is performed based on measurement data to improve the accuracy of the FE approximation and model order reduction (MOR) techniques are applied to determine a sufficiently low order system representation. A mixed-integer optimization problem is formulated and solved making use of different formulations of the observability Gramian to determine the optimal sensor locations and a Kalman filter is designed as an estimator based on a reduced order model. Moreover, a linear-quadratic regulator with integral part combined with the Kalman filter is developed to react efficiently towards disturbances. Finally this theoretical framework is tested in a real experiment.

Nonlinear finite element free and forced vibrations of cellular plates having lattice-type metamaterial cores: a strain gradient plate model approach

The main objective of this paper is to develop a theoretically and numerically reliable and efficient methodology based on combining a finite element method and a strain gradient shear deformation plate model accounting for the nonlinear free and forced vibrations of cellular plates having lattice-type metamaterial cores. The proposed model provides a modelling framework computationally more efficient than framework of the conventional three-dimensional elasticity for the simulation of the underlying nonlinear dynamics of cellular plates with microarchitectures. The corresponding governing equations follow Mindlin’s strain gradient elasticity theory including the micro-inertia effect applied to the first-order shear deformation plate theory along with the nonlinear von Kármán kinematics. Standard and higher-order computational homogenization methods determine the classical and strain gradient material constants, respectively. A higher-order C1-continuous 6-node finite element is adopted for the discretization in the spatial domain, and an arc-length continuation technique along with time-periodic discretization is implemented to solve the resulting nonlinear time-dependent problem. Through a set of comparative studies with 3D full-field models as reliable validation references for triangularly prismatic example cores, the accuracy and efficacy of the proposed dimension reduction methodology are demonstrated for a diverse range of problem parameters for analyzing the large-amplitude dynamic structural response. The results show that the proposed strain gradient plate model can accurately capture the microarchitecture effects of cellular plates with low computational costs, which signifies a relevant contribution particularly to the computational analysis of nonlinear dynamics.

Free Vibration and Buckling Analysis of Porous Two-Directional Functionally Graded Beams Using a Higher-Order Finite Element Model

Free Vibration and Buckling Analysis of Porous Two-Directional Functionally Graded Beams Using a Higher-Order Finite Element Model

High-Order Mixed Finite Element Variable Eddington Factor Methods

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the discrete ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory (LLNL). Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations.

A New 3D Marine Controlled-Source Electromagnetic Modeling Algorithm Based on Two New Types of Quadratic Edge Elements

This study proposes a new algorithm for a higher-order vector finite element method based on two new types of second-order edge elements to solve the electromagnetic field diffusion problem in a 3D anisotropic medium. To avoid source singularity in the quasistatic variant of Maxwell’s function, a secondary field formulation was adopted. The modeling domain was discretized using two types of quadratic edge hexahedral elements, which were obtained using the edge unification method to reduce variables on each side of two conventional quadratic edge elements. Compared with the traditional quadratic element, the number of unknowns that needed to be solved was significantly reduced. The sparse linear equation of the finite element system was solved using an open-source direct solver called MUMPS. The numerical results demonstrated that the proposed algorithm has the same level of accuracy as the conventional vector finite element method and has a significant advantage over it in terms of computational cost.

Selected aspects of constrained conforming approximation in higher-order FEM

This paper responds to numerous questions and inquiries related to constrained approximation (approximation with hanging nodes) in higher-order finite element methods that we received during the ESCO 2022 conference. We discuss explicit constraints enforced on linear algebra level as well as implicit constraints which are built into the basis functions of the finite element space, and selected additional mathematical and algorithmic aspects of conforming higher-order finite element approximations with one-level and arbitrary-level hanging nodes.

Stochastic Higher-order Finite Element Model for the Free Vibration of a Continuous Beam resting on Elastic Support with Uncertain Elastic Modulus

This paper deals with a continuous beam resting on elastic support with elastic modulus derived from a random process. Governing equations of the stochastic higher-order finite element method of the free vibration of the continuous beam were derived from Hamilton's principle. The random process of elastic modulus was discretized by averaging random variables in each element. A solution for the stochastic eigenvalue problem for the free vibration of the continuous beam was obtained by using the perturbation technique, in conjunction with the finite element method. Spectral representation was used to generate a random process and employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.

Heat and mass transfer characteristics in MHD Casson fluid flow over a cylinder in a wavy channel: Higher-order FEM computations

In this paper, an extensive analysis of heat and mass transfer characteristics of a Casson fluid flow past a cylinder in a wavy channel has been performed. Due to the non-availability of analytical solution the model represented by coupled non-linear partial differential equations has been simulated by implementing higher-order Finite Element Method (FEM). For the approximation of velocity, temperature, and concentration profiles, a finite element space involving the cubic polynomial (P3) is selected whereas the pressure estimation is accomplished through a space of quadratic polynomial (P2). Such a choice gives rise to 10° of freedom for each of the velocity components, temperature, and concentration profile while 6° of freedom for the pressure. The Newton's method is used to linearize the systems of equations. The linearized inner system of equations are solved with a direct solver PARDISO. Computational results are demonstrated in the form of velocity, isotherms, isoconcentrations, and for several quantities of interest including the drag and lift coefficients. It has been observed that both drag and lift coefficients show a decreasing trend with Magnetic parater M. In addition, for all values of Bn, the drag coefficient has a linear growth profile as a result of fluid forces dominating at higher values of Bn. Furthermore, the concentration is high in the center of the channel and has a decreasing trend as we move towards the walls.

Numerical analysis of a singularly perturbed convection diffusion problem with shift in space

We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.

A high-order finite element continuation for buckling analysis of porous FGM plates

This work presents an effective approach to investigating the buckling and post-buckling behavior of porous FGM plates. The main objective of the present analysis is to use a High Order Continuation based on the Asymptotic Numerical Method with the Finite Element Method for nonlinear behaviors of a Porous Functionally Graded Material plate (PFGM) with different porosity distributions under various types of transverse loads. In our study, we will take into account the lack of uniform and non-uniform porosity using a Higher-order Shear Deformation Theory (HSDT). The adapted solver proves to be more effective for analyzing non-linear mechanical phenomena and more precisely the post-buckling analysis of the PFGM plates. Indeed, this numerical high order algorithm presents an adaptive step length to the local nonlinearity of the solution branch and to detect easily the bifurcation points and bifurcated branches. When compared to iterative techniques like Newton–Raphson, the proposed high-order algorithm has the benefit of speeding up the resolution of nonlinear problems. The mathematical formulation is based on the higher-order shear deformation plate theory taking into consideration geometrical nonlinearity and initial imperfections. Numerical results of a parametric study on the effect of various types of load, porosity distribution, grading index, porosity index and different span-to-thickness ratios will be investigated.

Thermoelectric nonlinear vibration and buckling analysis of the smart porous core sandwich plate (SPCSP) resting on the elastic foundation

This paper analyses the vibration and buckling responses analysis of the smart porous core sandwich plate (SPCSP) resting on the elastic foundation under thermoelectric and thermomechanical loading. The plate is a constraint with conventional and unconventional boundary conditions. The thermo-mechanical properties of the plate are graded through the thickness direction with the sigmoid and exponential laws. The sandwich plate considered geometrical nonlinearity and assumed quadratic electric potential function across the thickness direction. The Hamilton principle derives the formulation with the first-order shear deformation theory (FSDT) displacement field. A nine-node quadratic element-based higher-order finite element method with seven degrees of freedom (DOFs) is used to convert the governing equation into a set of algebraic forms and solve it computationally (using MATLAB) with a modified Newton-Raphson scheme. Buckling and vibration responses are analysed under varying plate parameters like material exponent N, thickness ratio (a/h), porous exponent with different porosity dispersion, conventional and unconventional conditions, applied electric voltage V and temperature change [Formula: see text]. It is found that the presence of the Pasternak parameter gives additional stability to structures and overcomes the instability due to porosity. Also, the proposed work is not only precise but also simple in predicting the vibration and buckling response of smart sandwich porous plates under thermo-electric mechanical loading.

Hybrid finite element / multipole expansion method for atomic Kohn-Sham density functional theory calculations

A numerical framework is developed for aspherical atomic Kohn-Sham density functional theory calculations. The framework invokes higher-order finite elements as a radial discretization in combination with a multipole expansion for controlling the spherical resolution. Both all-electron and nonlocal pseudopotential calculations are addressed in a unified setting. The overall approach is validated through a range of numerical examples which demonstrate the systematic convergence of the radial and spherical discretizations as well as the outstanding accuracy that can be efficiently obtained in the presence of strong aspherical external fields. Overall, the presented approach offers a route to adaptive local enrichment for electronic structure calculation in the context of the finite element method.

Optimal local truncation error method for solution of 2-D elastodynamics problems with irregular interfaces and unfitted Cartesian meshes as well as for post-processing

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes recently developed for the scalar wave and heat equations for heterogeneous materials is extended to a more complex case of a system of the elastodynamics PDEs. Compact 9-point stencils (similar to those for linear finite elements) are used for OLTEM. Compared to our previous results, a new approach is used for the calculation of the right-hand side of the stencil equations due to body forces. It significantly simplifies the analytical derivations of OLTEM for time-dependent problems. There are no unknowns on interfaces between different materials; the structure of the global semi-discrete equations for OLTEM is the same for homogeneous and heterogeneous materials. For the first time we have also developed OLTEM with the diagonal mass matrix. In contrast to many known approaches with some ad-hoc calculations of the diagonal mass matrix, OLTEM offers a rigorous approach which is a particular case of OLTEM with the non-diagonal mass matrix. Another novelty of the article is a new post-processing procedure for the accurate calculations of stresses. It includes the same compact 9-point stencils as those in basic computations and uses the accelerations and the displacements at the grid points along with the PDEs for the stress calculations. OLTEM yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Numerical experiments for elastic heterogeneous materials show: a) at the same number of degrees of freedom (dof), OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; at the engineering accuracy of 0.1 % for the displacements, OLTEM reduces the number of dof by more than 20 times; at the engineering accuracy of 0.1 % for the stresses, OLTEM with the new post-processing procedure reduces the number of dof by more than 104 times compared to linear finite elements; b) at the same number of dof, OLTEM with unfitted Cartesian meshes is even more computationally efficient than high-order finite elements with much wider stencils and conformed meshes. This will lead to a huge reduction in the computation time for elastodynamics problems solved by OLTEM and will allow the direct computations of some complex wave propagation and structural dynamics problems for heterogeneous materials without the scale separation.

PROGRESSIVE DAMAGE ANALYSIS OF STEEL-REINFORCED CONCRETE BEAMS USING HIGHER-ORDER 1D FINITE ELEMENTS

The present work investigates progressive damage in steel-reinforced concrete structures. An elastic-perfectly plastic material response is considered for the reinforcing steel constituent, while the smeared-crack approach is applied to model the nonlinear behavior of concrete. The analysis employs one-dimensional numerical models based on higher-order finite elements derived using the Carrera unified formulation (CUF). A set of numerical assessments is presented to study the mechanical response of a steel-reinforced notched concrete beam loaded in tension. The predictions are found to be in very good agreement with reference experimental observations, thereby validating the numerical approach. It is shown that CUF allows for the explicit representation of the constituents within the composite beam, resulting in accurate solutions in a computationally efficient manner.

A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

<abstract><p>We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-$ 1_\sigma $ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in $ H^1 $ norm and $ L^2 $ norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant $ r $, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order $ \mathcal{O}(h^{k+1}+M^{-2}) $ in the $ L^\infty(L^2) $-norm has been established. We give several numerical examples to confirm the theory presented in this work.</p></abstract>