Finite Element Shape Functions Research Articles

Multiscale design of nonlinear materials using a Eulerian shape optimization scheme

AbstractMotivated by recent advances in manufacturing, the design of materials is the focal point of interest in the material research community. One of the critical challenges in this field is finding optimal material microstructure for a desired macroscopic response. This work presents a computational method for the mesoscale‐level design of particulate composites for an optimal macroscale‐level response. The method relies on a custom shape optimization scheme to find the extrema of a nonlinear cost function subject to a set of constraints. Three key “modules” constitute the method: multiscale modeling, sensitivity analysis, and optimization. Multiscale modeling relies on a classical homogenization method and a nonlinear NURBS‐based generalized finite element scheme to efficiently and accurately compute the structural response of particulate composites using a nonconformal discretization. A three‐parameter isotropic damage law is used to model microstructure‐level failure. An analytical sensitivity method is developed to compute the derivatives of the cost/constraint functions with respect to the design variables that control the microstructure's geometry. The derivation uncovers subtle but essential new terms contributing to the sensitivity of finite element shape functions and their spatial derivatives. Several structural problems are solved to demonstrate the applicability, performance, and accuracy of the method for the design of particulate composites with a desired macroscopic nonlinear stress‐strain response.

Why neural networks apply to scientific computing?

In recent years, neural networks have become an increasingly powerful tool in scientific computing. The universal approximation theorem asserts that a neural network may be constructed to approximate any given continuous function at desired accuracy. The backpropagation algorithm further allows efficient optimization of the parameters in training a neural network. Powered by GPU's, effective computations for scientific and engineering problems are thereby enabled. In addition, we show that finite element shape functions may also be approximated by neural networks.

A hierarchically enhanced recovery-based error estimator for bidimensional elastoplastic modelling by generalized finite element approach

This paper aims to contribute with a hierarchically enhanced recovery-based error estimator by using the Zienkiewicz–Zhu (ZZ) recovery procedure and a generalized finite element formulation. The proposed stress recovery consists in an approximation procedure that is enriched hierarchically by using partition of unity (PU) and an enrichment function. In other words, the approximation matrix used for stress field recovery employs the enriched finite element (FE) shape function. Furthermore, the edge-bubble hierarchically enriched strategy in the context of a generalized finite element approach is adopted to develop the enriched $$\hbox {C}^{0}$$ element and approximation the matrix. The error estimator based on this hierarchically enhanced recovery procedure and the generalized finite element approach is employed to solve several elastic and elastoplastic hardening benchmark problems. Moreover, the effectivity index, refinement index, recovered stress field, global error and local error, both in energy norm, are evaluated in several applications. In addition, the sensitivity of the proposed recovery procedure to a severely distorted mesh is also analysed. The results obtained by the proposed procedure is compared to other well-established recovery procedures and finite element approaches. By comparing the results, the robustness in the numerical performance, the versatility in the computational implementation and the competitivity of the proposed procedure are demonstrated in the applications.

A vibration based acoustic wave propagation technique for assessment of crack and corrosion induced damage in concrete structures

Early detection of small concrete crack or reinforcement corrosion is necessary for Structural Health Monitoring (SHM). Global vibration based methods are advantageous over local methods because of simple equipment installation and cost efficiency. Among vibration based techniques, FRF based methods are preferred over modal based methods. In this study, a new coupled method using frequency response function (FRF) and proper orthogonal modes (POM) is proposed by using the dynamic characteristic of a damaged beam. For the numerical simulation, wave finite element (WFE), coupled with traditional finite element (FE) method is used for effectively incorporating the damage related information and faster computation. As reported in literature, hybrid combination of wave function based wave finite element method and shape function based finite element method can addresses the mid frequency modelling difficulty as it utilises the advantages of both the methods. It also reduces the dynamic matrix dimension. The algorithms are implemented on a three-dimensional reinforced concrete beam. Damage is modelled and studied for two scenarios, i.e., crack in concrete and rebar corrosion. Single and multiple damage locations with different damage length are also considered. The proposed methodology is found to be very sensitive to both single- and multiple- damage while being computationally efficient at the same time. It is observed that the detection of damage due to corrosion is more challenging than that of concrete crack. The similarity index obtained from the damage parameters shows that it can be a very effective indicator for appropriately indicating initiation of damage in concrete structure in the form of spread corrosion or invisible crack.

An enriched finite element/level-set method for simulating two-phase incompressible fluid flows with surface tension

A finite element method is introduced to simulate surface tension dominated flow of two immiscible fluids featuring an enriched space for capturing both strong and weak pressure discontinuities. The proposed enriched finite element space is created utilizing the standard finite element shape functions. Discontinuities are captured by adding merely one additional degree of freedom per each node of the elements cut by the interface. Being local to the cut elements, these additional degrees of freedom are eliminated before assembling the global system of equations following a condensation procedure. The method is stabilized introducing a procedure for improving the conditioning of the enriched pressure contribution to the stiffness matrix in small-cut situations. An improved smoothing strategy based on an artificial diffusion equation is proposed to enhance the performance of the method on rather coarse meshes. A series of three-dimensional two-phase fluid flow benchmarks are solved to assess the performance of the method. Particular attention is paid to surface tension dominated cases. The method is verified by showing its accuracy in capturing strong pressure discontinuity at the interface of a spherical droplet as well as its capability in handling large pressure gradient discontinuity in a hydrostatic liquid–gas container. The method is further validated by simulating oscillations of a slightly disturbed spherical droplet. The mass conservation property of the method and the effect of the smoothing procedure on the result is assessed by simulating the oscillations of a prolate droplet. Ultimately, the method is tested in a more challenging setting by simulating the rising gas bubble inside a liquid domain.

Enriched finite element modeling in the dynamic analysis of plane frame subject to random loads

This work contributes to the generalized finite element approach in free vibration, dynamic elastic, and elastoplastic analysis of plane frame subjected to random excitation generated by the wind action. The wind velocity is modeled mathematically by using power spectral density method in combination with Shinozuka’s model, along with the commonly employed wind spectra. From these spectra, the dynamic wind loading is determined from the sum of the mean and floating wind velocities. The governing equation is formulated by Euler–Bernoulli beam theory, and it is discretized by using the enriched beam element. The enrichment is done by employing enriched finite element shape function to construct the enriched mathematical space. This strategy is constituted by the enrichment space, which is constructed by trigonometric functions, and the conventional space, which is constructed by conventional two-node Lagrange–Hermite shape function. The time increment procedure is carried out by Hilber-Hughes-Taylor algorithm and the material nonlinearity is modeled by von Mises isotropic hardening model, solved by the Newton–Raphson algorithm. A flowchart is presented to summarize the proposed numerical modeling procedure. Finally, several applications are presented, and the results obtained by the generalized finite element method are compared with those obtained by conventional beam element. Natural frequencies are determined in a one-story plane frame and are compared with reference results. The relative error in displacement is determined in h-refine strategy for quadratic beam element (FEM3), while the generalized finite element method adopts the enrichment increment strategy. The results demonstrate the competitiveness and numerical stability of generalized finite element method in this type of application. Even in comparison to the quadratic beam element, the generalized finite element method presents good performance and accuracy in numerical modeling.

DOF‐gathering stable generalized finite element methods for crack problems

AbstractGeneralized or eXtended finite element methods (GFEM/XFEM) have been studied extensively for crack problems. Most of the studies were concentrated on localized enrichment schemes where nodes around the crack tip are enriched by products of singular and finite element shape functions. To attain the optimal convergence rate O(h) (h is the mesh‐size), nodes in a fixed domain containing the tip have to be enriched. This results in many extra degrees of freedom (DOF) and stability issues. A so‐called DOF‐gathering GFEM/XFEM can avoid the increase of DOF, by collecting the singular enriched DOF together. Various novel modifications were designed for the DOF‐gathering GFEM/XFEM to get the optimal convergence O(h). However, they could not improve the stability, namely, condition numbers of stiffness matrices of the DOF‐gathering GFEM/XFEM could be much larger than that of the standard FEM. Motivated from the idea of stable GFEM, we propose in this paper a DOF‐gathering stable GFEM (d.g.SGFEM) for the Poisson problem with crack singularities. The main idea is to modify the singular and Heaviside enrichments by subtracting their finite element interpolants. The optimal convergence O(h) of the proposed d.g.SGFEM is proven theoretically. Moreover, the condition number of stiffness matrices of d.g.SGFEM, utilizing a local orthgonalization technique, is shown to be of same order as that of the standard FEM. Two kinds of commonly used cut‐off functions used to gather the DOF are analyzed in a unified approach. Theoretical convergence and the conditioning results of d.g.SGFEM are verified by numerical experiments.

Multidimensional Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics

We present the second-order multidimensional Staggered Grid Hydrodynamics Residual Distribution (SGH RD) scheme for Lagrangian hydrodynamics. The SGH RD scheme is based on the staggered finite element discretizations as in [Dobrev et al., SISC, 2012]. However, the advantage of the residual formulation over classical FEM approaches consists in the natural mass matrix diagonalization which allows one to avoid the solution of the linear system with the global sparse mass matrix while retaining the desired order of accuracy. This is achieved by using Bernstein polynomials as finite element shape functions and coupling the space discretization with the deferred correction type timestepping method. Moreover, it can be shown that for the Lagrangian formulation written in non-conservative form, our residual distribution scheme ensures the exact conservation of the mass, momentum and total energy. In this paper we also discuss construction of numerical viscosity approximations for the SGH RD scheme allowing to reduce the dissipation of the numerical solution. Thanks to the generic formulation of the staggered grid residual distribution scheme, it can be directly applied to both single- and multimaterial and multiphase models. Finally, we demonstrate computational results obtained with the proposed residual distribution scheme for several challenging test problems.

Применение сингулярных финитных функций для аппроксимации поверхностной плотности заряда в методе граничных интегральных уравнений

In this paper we consider application of singular functions for constructing finite element shapefunctions in problems of the electric field computation by the method of secondary sources. The main idea of this approach is based on the use of analytical solutions of the Laplace equation in the vicinity of special points of charged electrodes. Compared to traditional technologies of the charge density approximation on electrode surfaces, the considered approach allows to achieve high accuracy of calculation of electric field intensity in the vicinity of the charged object, including its angles and faces. The efficiency and high accuracy of the results obtained with such approximations is approved by comparison of the numerically derived field distributions with available analytical solutions.

A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements

The Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but small strains and deformations. The FFRF is conventionally derived via a continuum mechanics approach. This tedious and circuitous approach, which still attracts attention among researchers, yields so-called inertia shape integrals. These unhandy volume integrals, arising in the FFRF mass matrix and quadratic velocity vector, depend not only on the degrees of freedom, but also on the finite element shape functions. That is why conventional computer implementations of the FFRF are laborious and error prone; they require access to the algorithmic level of the underlying finite element code or are restricted to a lumped mass approximation. This contribution presents a nodal-based treatment of the FFRF to bypass these integrals. Each flexible body is considered in its spatially discretized state ab initio, wherefore the integrals are replaced by multiplications by a constant finite element mass matrix. Besides that, this approach leads to a simpler and concise but rigorous derivation of the equations of motion. The steps to obtain the inertia-integral-free equations of motion (in 2D and 3D spaces) are presented in a clear and comprehensive way; the final result provides ready-to-implement equations of motion without a lumped mass approximation, in contrast to the conventional formulation.

A Novel Triangular Element with Continuous Nodal Acoustic Pressure Gradient for Acoustic Scattering Problems

To improve the performance of the standard finite element (FE) method in acoustic simulation, a novel triangular element with continuous nodal acoustic pressure gradient (FEM-T3-CNG) is presented to solve two-dimensional underwater acoustic scattering problems. In this FEM-T3-CNG model, the local approximation (LA) is represented by using the least-squares (LS) scheme, and the standard FE shape functions are employed to satisfy the partition of unity (PU) concept. In order to possess the important delta Kronecker property, the constrained orthonormalized LS (CO-LS) is utilized to construct the hybrid nodal shape functions. Incorporating the present FEM-T3-CNG element with the proper nonreflecting boundary condition, the two-dimensional underwater acoustic scattering problems in the infinite domain could be solved ultimately. The numerical results show that the present FEM-T3-CNG element behaves much better than the standard FEM-T3 element in terms of computation accuracy and can be regarded as a good alternative approach in exterior acoustic computation.

Flow of non‐Newtonian fluids in fractured porous media: Isogeometric vs standard finite element discretisation

SummaryA formulation has been derived for the flow of non‐Newtonian (power‐law) fluids in deformable, fractured porous media. The formulation is enhanced with a subgrid scale model to accurately represent the flow of the power‐law fluids inside the cracks. The resulting equations have been discretised using standard (Lagrangian) finite element shape functions and with non‐uniform rational B‐splines (NURBS), which have been cast into a standard finite element datastructure using Bézier extraction. The effect of the power‐law index on the velocity inside the fracture and on the total fluid flow through the porous medium has been analysed for a typical boundary‐value problem. It is shown that large differences between non‐Newtonian and linearised Newtonian fluids can occur for the fluid velocity inside the fracture. This can significantly influence the total fluid transport through the domain. A mesh sensitivity study has been carried out as well and shows that markedly smaller element sizes are required in order to obtain accurate results for the local flow inside the fracture, compared with the element sizes necessary for obtaining accurate results inside the porous medium away from the fracture. Moreover, a comparison has been made between the results obtained using standard Lagrange polynomials and those obtained using NURBS. It is shown that while both discretisation methods are able to accurately simulate the deformations and pressures in the porous medium, the higher interelement continuity of NURBS is mandatory for obtaining correct values of the fluid velocities inside the fracture, especially near the tips.

Fully numerical Hartree‐Fock and density functional calculations. I. Atoms

AbstractAlthough many programs have been published for fully numerical Hartree‐Fock (HF) or density functional (DF) calculations on atoms, we are not aware of any programs that support hybrid DFs, which are popular within the quantum chemistry community due to their better accuracy for many applications, or that can be used to calculate electric properties. Here, we present a variational atomic finite element solver in the HelFEM program suite that overcomes these limitations. A basis set of the type is used, where B n(r) are finite element shape functions and are spherical harmonics, which allows for an arbitrary level of accuracy. HelFEM supports nonrelativistic HF and DF calculations even with hybrid functionals, which are not available in other commonly available program packages. Hundreds of functionals at the local spin‐density approximation (LDA), generalized‐gradient approximation (GGA), as well as the meta‐GGA levels of theory are included through an interface with the Libxc library. Electric response properties are achievable via finite field calculations. We introduce an alternative grid that yields faster convergence to the complete basis set limit than commonly used alternatives. We also show that high‐order Lagrange interpolating polynomials yield the best convergence, and that excellent agreement with literature HF limit values for electric properties, such as static dipole polarizabilities, can be achieved with the present approach. Dipole moments and dipole polarizabilities at finite field are also reported with the PBE, PBEh, TPSS, and TPSSh functionals. Finally, we show that a recently published Gaussian basis set is able to reproduce absolute HF and DF energies of neutral atoms, cations, as well as anions within a few dozen microhartrees.

Fully numerical Hartree‐Fock and density functional calculations. II. Diatomic molecules

AbstractWe present the implementation of a variational finite element solver in the HelFEM program for benchmark calculations on diatomic systems. A basis set of the form is used, where (μ, ν, φ) are transformed prolate spheroidal coordinates, B n(μ) are finite element shape functions, and are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree‐Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin density approximation (LDA), generalized gradient approximation (GGA), and the meta‐GGA level can be used through an interface with the Libxc library; meta‐GGA and hybrid DFs are not available in other fully numerical diatomic program packages. Finite electric fields are also supported in HelFEM, enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HelFEM and the novel basis set procedure are demonstrated by reproducing the restricted open‐shell HF limit energies of 68 diatomic molecules from the first to the fourth period with excellent agreement with literature values, despite requiring orders of magnitude fewer parameters for the wave function. Then, the electric properties of the BH and N2 molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed for the literature references. Finally, HF, LDA, GGA, and meta‐GGA calculations of the atomization energy of N2 are performed, demonstrating the superb accuracy of the present approach.

A mixed B-bar formulation derived by a principle of virtual power for energy–momentum time integrations of fiber-reinforced continua

The optimization of finite element discretizations within energy–momentum time integrations of thermodynamical problems leads to further ‘constraints’ on algorithmic modifications compared to static problems, because each modification must not violate discrete balance laws. In order to preserve each discrete balance law while modifying the space approximation, a mixed version of the principle of virtual power is applied. Besides the reduction of volumetric locking in the matrix material and line locking in the fibers of the considered transversely isotropic continuum, a new space approximation additionally prevents spurious shear deformations in bending dominated problems. This approximation is based on independent fields for the deformation gradient and the first Piola–Kirchhoff stress tensor, leading to a new B-bar operator. The local shape functions for the independent deformation gradient are derived by a new criterion from the finite element shape functions. In this way, linear and quadratic hexahedral finite elements are combined with standard and non-standard shape functions of tetrahedral and prismatic elements. In order to evaluate the performance of this new mixed B-bar method with respect to locking as well as thermodynamical behavior, numerical examples with thin-walled, fiber-reinforced structures are considered. The numerical examples also reveal a new aspect of structure preservation, namely, the goal of a spatial mesh density independent structure preservation.

Convergence and error analysis of FE-HMM/FE[formula omitted] for energetically consistent micro-coupling conditions in linear elastic solids

A cornerstone of numerical homogenization is the equivalence of the microscopic and the macroscopic energy densities, which is referred to as Hill–Mandel condition. Among these coupling conditions, the cases of periodic, linear displacement and constant traction conditions are most prominent in engineering applications. While the stiffness hierarchy of these coupling conditions is a theoretically established and numerically verified result, very little is known about the numerical errors and convergence properties for each of them in various norms. The present work addresses these aspects both on the macroscale and the microscale for linear as well as quadratic finite element shape functions. The analysis addresses aspects of (i) regularity and how its loss affects the convergence behavior on both scales compared with the a priori estimates, of (ii) error propagation from micro to macro and of (iii) optimal micro–macro mesh refinement strategy. For constant traction conditions two different approaches are compared. The performance of a recovery-type error estimation based on superconvergence is assessed. All results of the present work are valid for both the Finite Element Heterogeneous Multiscale Method FE-HMM and for FE2.

Wing Crack Propagation Simulation Using Scaled Boundary Finite Element Method

Considering the existence of side-face loads, a new scaled boundary finite element (SBFEM) shape function is used to study the problem of wing crack propagation. In the solution of crack contact, the contact interface constraint is introduced by Lagrange multiplier method. The polygon SBFEM remeshing technology is used to simulate the crack propagation problem with single and multiple wing crack. Compared with related experimental and numerical simulation results, the correctness and effectiveness of the new SBFEM shape function in solving the wing crack propagation are verified. It is found that under pressure, the wing crack propagates along the direction of the maximum principal compressive stress during initial expansion, and then the crack opens and gradually expands toward the pressure. Wing cracks only have contact phenomena on the initial crack surface during crack propagation. The contact force gradually decreases from the centre to the sides of the initial crack. The friction coefficient has a great influence on the crack propagation direction.

Multifield Variational Sectional Analysis for Accurate Stress Computation of Multilayered Composite Beams

A multifield variational beam sectional analysis is proposed based on the Hellinger–Reissner principle for anisotropic beams with arbitrary cross-sectional geometries and material distributions. Both stresses and three-dimensional (3D) warping deformations are treated as unknowns, which are modeled using the isoparametric finite element (FE) shape functions. The nodal stresses are solved on each material domain, resulting in the required continuity within the domain and discontinuity at the material interfaces. A generalized Timoshenko-like stiffness matrix is constituted from the 3D warping solution inherently describing Timoshenko model for transverse shear, Poisson deformations, and associated couplings. The accuracy of the present analysis is demonstrated for benchmark composite beams with elastic couplings. The computed elastostatic responses are in excellent agreement with those of 3D FE analysis and experimental data. The stress predictions from the present multifield formulation show excellent correlation with 3D FE results using fewer elements in comparison to the conventional displacement-based approach while maintaining the stress continuity requirements for multilayered composite beams.

Implementation aspects of a phase-field approach for brittle fracture

This paper provides a comprehensive overview of a phase-field model of fracture in solid mechanics setting. We start reviewing the potential energy governing the whole process of fracture including crack initiation, branching or merging. Then, a discretization of system of equation is derived, in which the key aspect is that for the correctness of fracture phenomena, a split into tensile and compressive terms of the strain energy is performed, which allows crack to occur in tension, not in compression. For numerical analysis, standard finite element shape functions are used for both primary fields including displacements and phase field. A staggered scheme which solves the two fields of the problem separately is utilized for solution step and illustrated with a segment of Python code.

Scaled boundary finite element method with exact defining curves for two-dimensional linear multi-field media

This paper presents an efficient and accurate numerical technique based upon the scaled boundary finite element method for the analysis of two-dimensional, linear, second-order, boundary value problems with a domain completely described by a circular defining curve. The scaled boundary finite element formulation is established in a general framework allowing single-field and multi-field problems, bounded and unbounded bodies, distributed body source, and general boundary conditions to be treated in a unified fashion. The conventional polar coordinates together with a properly selected scaling center are utilized to achieve the exact description of the circular defining curve, exact geometry of the domain, and exact spatial differential operators. Standard finite element shape functions are employed in the discretization of both trial and test functions in the circumferential direction and the resulting eigenproblem is solved by a selected efficient algorithm. The computational performance of the implemented procedure is then fully investigated for various scenarios to demonstrate the accuracy in comparison with standard linear elements.