Finite Element Interpolation Research Articles

Simpson’s Variational Integrator for Systems with Quadratic Lagrangians

This contribution proposes a variational symplectic integrator aimed at linear systems issued from the least action principle. An internal quadratic finite-element interpolation of the state is performed at each time step. Then, the action is approximated by Simpson’s quadrature formula. The implemented scheme is implicit, symplectic, and conditionally stable. It is applied to the time integration of systems with quadratic Lagrangians. The example of the linearized double pendulum is treated. Our method is compared with Newmark’s variational integrator. The exact solution of the linearized double pendulum example is used for benchmarking. Simulation results illustrate the precision and convergence of the proposed integrator.

A fast Bayesian parallel solution framework for large-scale parameter estimation of 3D inverse heat transfer problems

The inverse heat transfer problems (IHTP) have a wide range of applications in the engineering field. Bayesian methods using Markov Chain Monte Carlo (MCMC) have long been considered as a robust and effective method for solving inverse problems. However, the discretization of the problem domain by the spatio-temporal Galerkin skill, i.e., the finite element interpolation also includes the time dimension, making the scale of the unknown parameters extremely difficult for Bayesian calculations. In this paper, a fast Bayesian parallel sampling (FBPS) framework is proposed for large-scale parameter estimation of benchmark three-dimensional inverse heat transfer problems (3D-IHTP). The FBPS we developed achieves a parameter computation scale of 105 magnitude within minutes, through dimensionality reduction of the space-time dependent problem domain. The Hamiltonian Monte Carlo (HMC) sampler, which is proven to be more efficient for high-dimensional parameter estimation, is employed. Through several simulation tests of IHTP, it was confirmed that the solving efficiency of FBPS surpasses that of the traditional Bayesian strategy significantly. Finally, FBPS is successfully developed to estimate the unknown heat flux on the chip heat sink and pack interface effectively, given some simulated high resolution measurement data. The reliability and efficiency show that FBPS has the potential to support efficient prediction techniques for a class of IHTPs in engineering applications.

Free vibration analysis of sandwich structure via point interpolation meshfree method based on polynomial basic function

The stability of circular and annular sector plate structures under aerodynamic pressure in supersonic conditions is crucial for aircraft components. Conventional uniform materials lead to heavy structures, which are undesirable for aircraft. This study explores the use of functionally graded materials to enhance flutter stability. It employs a refined logarithmic shear deformation theory to describe plate displacement, classical elasticity theory for constitutive equations, and Hamilton’s principle for equations of motion. Two methods, finite element, and Meshfree radial point interpolation, are used to solve these equations and verify the results. Parameter studies are conducted to assess the effects of grading index, Biot’s coefficient, span angle, and geometrical aspect ratio, revealing unexpected adverse effects of grading material on stability.

A penalty-free Shifted Boundary Method of arbitrary order

We introduce and analyze a penalty-free formulation of the Shifted Boundary Method (SBM), inspired by the asymmetric version of the Nitsche method. We prove its stability and convergence for arbitrary order finite element interpolation spaces and we test its performance with a number of numerical experiments. Moreover, while the SBM was previously believed to be only asymptotically consistent (in the sense of Galerkin orthogonality), we prove here that it is indeed exactly consistent.

Use of XTFEM based on the consecutive interpolation procedure of quadrilateral element to calculate J-integral and SIFs of an FGM plate

An extended twice - interpolation finite element method (XTFEM) based on the consecutive interpolation procedure (CIP) of quadrilateral element in two-dimensional (2D) with continuous nodal stress for stress fields near material interface and geometry boundary such as inclusion and hole is presented. In contrast to the traditional approaches, the approximation shape functions constructed based on the CIP involve both nodal values and averaged nodal gradients as interpolation conditions. The CIP for quadrilateral element is developed recently by Tinh Quoc Bui, et al., 2014.In this research, we exhibit a pioneering extension of CIP – based consecutive-interpolation quadrilateral element enhanced by enrichment in terms of local partition of unity method to precisely model of a functionally graded material (FGM) plate had a single-edge crack a, and the influence of holes and inclusions near the crack was investigated in 2D structure to calculate the stress intensity factors (SIFs), an important parameter in predicting the direction of the crack even when the crack is no longer growing, taking advantages of the strengths and making use all the desirable features of both techniques, the CIP and the local enriched partition of unity method. Using MATLAB codes combined with XTFEM based on CIP of quadrilateral element. The accuracy and performance of the proposed XTFEM are compared to standard finite element method (FEM) (ANSYS). The comparison revealed the following analytical results: SIFs of the FGM isotropic plate had a crack a with an average error of 2.55% and the J-integral of 2.86%. The XTFEM gave better errors than FEM compared to the Exact and Experimental methods.

Sparse polynomial prediction

In numerical analysis, sparse grids are point configurations used in stochastic finite element approximation, numerical integration and interpolation. This paper is concerned with the construction of polynomial interpolator models in sparse grids. Our proposal stems from the fact that a sparse grid is an echelon design with a hierarchical structure that identifies a single model. We then formulate the model and show that it can be written using inclusion–exclusion formulæ. At this point, we deploy efficient methodologies from the algebraic literature that can simplify considerably the computations. The methodology uses Betti numbers to reduce the number of terms in the inclusion–exclusion while achieving the same result as with exhaustive formulæ.

Two families of two‐dimensional finite element interpolation functions of general degree with general continuity

AbstractThis work presents a generalization of the functions used in Fraeijs de Veubeke's quadrilateral (FVQ) element and in Hsieh, Clough and Tocher's (HCT) triangle, allowing for an arbitrary degree of the approximation, , and for arbitrary order of continuity, . These functions can be used to define finite elements that lead by direct assembly to approximations. Their determination for each element is procedural, generalizing the reasoning used by Zienkiewicz to explain the FVQ element. The completeness of each basis is confirmed by comparing its dimension with that of a broken basis, defined primitive element wise, upon which the continuity constraints are successively imposed. The order of continuity at the vertices that is required to enable the independent definition of the normal derivatives at the sides is confirmed numerically. The corresponding interpolation functions are symbolically derived using Mathematica, and simple matlab codes using the data thus obtained are made available, to illustrate their application.

Two-dimensional model for composite beams and its state-space based mixed finite element solution by DQM

Most one-dimensional composite beam theories are based on simplified shear deformation assumptions, or even ignore shear deformation, which have limitations in analyzing multilayer beams with significant material differences. Therefore, this paper firstly proposed a two-dimensional analytical model for composite beams through the equivalent transformation of cross section. Based on the mixed variational principle, the state equations are then derived by finite element meshing and interpolation along the length of the beam, with nodal displacements and their energy-conjugated stresses as state variables. Subsequently, the differential quadrature method (DQM) is introduced to solve the equations and a two-dimensional analysis method for composite beams is established. Due to the use of displacements and stresses as fundamental variables in the state equations, various transfer characteristics of displacements and stresses at the interlayer interfaces of composite beams can be conveniently handled. This method is finally verified by numerical examples of steel-concrete composite beams and concrete beams with corrugated steel webs. Since no assumptions of displacement and stress distributions along the thickness of beams are required, the present method can provide benchmarks for various one-dimensional beam theories.

Hybrid fundamental-solution-based FEM for thermal analysis of cellular materials containing non-smooth holes

A novel hybrid polygonal finite element is proposed for simulating the heat conduction in two-dimensional cellular materials based on the hybrid fundamental-solution-based finite element method (HFS-FEM). The proposed method assumes two independent temperature fields in the domain and boundary of the element, respectively. The intra-element temperature field is approximated by a combination of fundamental solutions satisfying the governing equation of the problem. The charge simulation method is introduced so that the required fundamental solution can be established for the domain with an arbitrarily-shaped hole. To satisfy the continuity condition between adjacent elements, a conforming frame temperature field is approximated by the conventional finite element interpolation functions. Combing two independent fields into the modified variational functional leads to the resultant finite element formulation avoiding the need for domain integration. Such boundary-type feature endows a great capability of constructing any polygonal element with different number of sides and makes a great flexibility in mesh generation. The accuracy and efficiency of HFS-FEM are demonstrated through three numerical examples which includes triangular, astroid and irregular hole elements.

Space-time hp-finite elements for heat evolution in laser powder bed fusion additive manufacturing

The direct numerical simulation of metal additive manufacturing processes such as laser powder bed fusion is challenging due to the vast differences in spatial and temporal scales. Classical approaches based on locally refined finite elements combined with time-stepping schemes can only address the spatial multi-scale nature and provide only limited scaling potential for massively parallel computations. We address these shortcomings in a space-time Galerkin framework where the finite element interpolation also includes the temporal dimension. In this setting, we construct four-dimensional meshes that are locally refined towards the laser spot and allow for varying temporal accuracy depending on the position in space. By splitting the mesh into conforming time-slabs, we recover a stepwise solution to solve the space-time problem locally in time at this slab; additionally, we can choose time-slab sizes significantly larger than classical time-stepping schemes. As a result, we believe this setting to be well suited for large-scale parallelization. In our work, we use a continuous Galerkin–Petrov formulation of the nonlinear heat equation with an apparent heat capacity model to account for the phase change. We validate our approach by computing the AMB2018-02 benchmark, where we obtain an excellent agreement with the measured melt pool shape. Using the same setup, we demonstrate the performance potential of our approach by hatching a square area with a laser path length of about one meter.

A locking-free nonconforming FEM for optimal control problems governed by linear elasticity equations

In this paper, we investigate a locking-free nonconforming finite element method (FEM) for linear elasticity optimal control problems (OCPs) via the variational control discretization approach. Due to the fact that the finite element interpolation operator is identical to the Ritz projection, optimal error estimates are established for the control, state and adjoint state variables, respectively. Numerical experiments are also provided to validate the theoretical results.

The high-order Shifted Boundary Method and its analysis

The Shifted Boundary Method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods, that has proven efficient in handling partial differential equation problems with complex geometries. The key feature of the SBM is a shift in the location where boundary conditions are applied – from the true to a surrogate boundary – and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. This paper presents the high-order version of the method, its mathematical analysis, and numerical experiments. The proposed method retains optimal accuracy for any order of the finite element interpolation spaces despite the surrogate boundary is piecewise linear. As such, the proposed approach bypasses the problematic issue of meshing complex geometries with high-order body-fitted boundary representations, without the need of complex data structures for the integration on cut cells.

Regional application of C1 finite element interpolation method in modeling of ionosphere total electron content over Europe

• A novel approach for regional ionosphere grid model with high spatial-temporal resolution is suggested. • The results of the new model is analyzed at different disturb geomagnetic and solar conditions. • A comparative analysis between the results of new model, GIM, NeQuick and GPS-TEC is made. In this paper, a novel approach based on C 1 finite element (FE) interpolation is used to construct a regional ionospheric grid model (RIGM) with high spatio-temporal resolution over Europe. The spatial resolution of the new model is 0.5 0 for a longitude and latitude and a temporal resolution is 15 min. Observations of 38 GPS stations at different days and seasons with different geomagnetic and solar activity conditions are used to evaluate the new method. All stations are used to discretize the whole domain of the problem into separate triangular patches using a Delaunay triangulation algorithm. In this case, GPS stations would be the vertices of the triangular patches. Also, the position of stations in sun-fixed coordinate system with coordinates of latitude and longitude are chosen in order to consider the temporal variations of vertical total electron content (VTEC). To model the behavior of VTEC in each triangular patch, a particular shape function is defined which is fitted to VTEC values at three vertices of each triangle while it satisfies C 1 continuity of VTEC at the common edges of the adjacent triangles. The results of the new model are compared with the VTEC of GPS at 4 control stations as well as global ionosphere map (GIM) and NeQuick models. Statistical parameters, including correlation coefficient, root mean square error (RMSE), standard deviation and dVTEC = |VTEC GPS - VTEC model | have been used to evaluate the error of the models. In low geomagnetic activity (KP < 4), the average RMSE at 4 control stations for FE, GIM and NeQuick models are 0.89, 1.05 and 6.42 TECU, respectively, while in the high geomagnetic activity (KP > 4), they are 1.24, 1.68 and 7.62 TECU, respectively. For further analysis, the accuracy of three models in precise point positioning (PPP) has been also evaluated. In PPP method, there is an improvement of about 1–16 mm compared to GIM and NeQuick models at coordinate components of control stations in high geomagnetic activity. The results of the analysis performed in this paper show that the FE model has a very high capability and accuracy in regional VTEC modeling in high and low geomagnetic conditions. The regional accuracy of this model is higher than GIM and NeQuick models and therefore can be considered as a new model for the ionosphere.

A computational framework for biomaterials containing three-dimensional random fiber networks based on the affine kinematics.

Understanding the structure-function relationship of biomaterials can provide insights into different diseases and advance numerous biomedical applications. This paper presents a finite element-based computational framework to model biomaterials containing a three-dimensional fiber network at the microscopic scale. The fiber network is synthetically generated by a random walk algorithm, which uses several random variables to control the fiber network topology such as fiber orientations and tortuosity. The geometric information of the generated fiber network is stored in an array-like data structure and incorporated into the nonlinear finite element formulation. The proposed computational framework adopts the affine fiber kinematics, based on which the fiber deformation can be expressed by the nodal displacement and the finite element interpolation functions using the isoparametric relationship. A variational approach is developed to linearize the total strain energy function and derive the nodal force residual and the stiffness matrix required by the finite element procedure. Four numerical examples are provided to demonstrate the capabilities of the proposed computational framework, including a numerical investigation about the relationship between the proposed method and a class of anisotropic material models, a set of synthetic examples to explore the influence of fiber locations on material local and global responses, a thorough mesh-sensitivity analysis about the impact of mesh size on various numerical results, and a detailed case study about the influence of material structures on the performance of eggshell-membrane-hydrogel composites. The proposed computational framework provides an efficient approach to investigate the structure-function relationship for biomaterials that follow the affine fiber kinematics.

Revisiting p-refinement in structural topology optimization

The results obtained in traditional topology optimization with a single density design variable assigned to each element, are typically improved through h-refinement where the resolution of the FE mesh is scaled up along with the number of density elements/voxels, without increasing the order of the finite element interpolation. Higher order elements allow a more realistic simulation of the physics without some of the spurious results associated with lower order elements, including improved numerical instability as a tertiary advantage. In this work, we conduct a detailed investigation of p-refinement in the context of SIMP topology optimization including optimized topologies, compliance values and CPU clock time, for various 2D classical benchmark problems.

Local refinement with arbitrary irregular meshes and implementation in numerical manifold method

The solution accuracy based on finite element interpolation strongly depends on mesh quality. Thus, the numerical manifold method advocates that if a finite element mesh is selected as the mathematical cover, the mesh should be uniform, where all elements are congruent squares or isosceles right triangles for two-dimensional problems. However, the elements close to the points of stress concentrations or singularities must be smaller than those further away from these points. This leads to an irregular mesh in which a few nodes of certain small elements are hung on the edges of adjacent larger elements. This study implements local refinement with arbitrary k-irregular meshes by selecting piecewise polynomials as the local approximations over physical patches and enforcing appropriate constraints on the piecewise polynomials. All the elements are ordinary isoparametric elements, and elements with variable numbers of nodes on edges are not required. No linear dependency issue exists. Furthermore, refinement with any number of mesh layers is developed so that solution error can be distributed as uniformly as possible. Typical examples with different singularities are designed to illustrate the effectiveness of the proposed procedure.

An evaluation of MITC and ANS elements in the nonlinear analysis of shell structures

The recently emerged idea of incorporating higher-order strain or displacement discontinuities into standard finite element interpolations has triggered the development of robust techniques, which allow efficient modeling of large deflections and rotations. A number of studies on efficient elements with embedded high-order discontinuities, including mixed interpolation tensorial components (MITC) and assumed natural strain (ANS) elements, were published during the past decade. It was verified that local enrichments of the displacement and/or strain interpolation can enhance the responses of displacements and stresses by efficient models. The multitude of methods proposed in the literature calls for a comparative study that would present the diverse approaches in a unified framework, point out their common features and differences, and find their limits of applicability. In this regard, by proposing two robust triangular elements based on the higher-order strain field, mixed and strain-based approaches are compared. The strengths and weaknesses of individual formulations are elucidated by analyzing the convergence behavior. After determining the optimal condition, the nonlinear theoretical analysis of functionally graded (FG) shells has been performed by employing the weighted arc-length method. The equilibrium path for load-bearing factor versus post-buckling capacity, the error rate path and the residual force norm are also presented in different material and boundary conditions.

Morphological simplification of asphaltic mixture components for micromechanical simulation using finite element method

AbstractIn micromechanical simulations of asphalt mixtures, complex shapes of asphalt mortar and coarse aggregates require numerous elements in the finite element (FE) representation. This makes simulation time‐consuming with a high risk of failure. To conduct simulations with acceptable accuracy and low computational cost, an approach is proposed to simplify the geometry of mixture microstructures with error assessment due to simplification. The methodology comprises four main steps, as follows: (1) coarse aggregates in a specimen are three‐dimensionally reconstructed to conduct surface triangulation; (2) surface triangles are then clustered with a predefined fitting accuracy using the least square method; (3) coarse aggregates are simplified as polyhedrons based on triangle clusters, and the asphalt mortar is then reconstructed using Boolean operations; and (4) the simulation error caused by simplification is assessed based on FE interpolation theory to ensure the reliability of the simulation. Two samples were prepared and then reconstructed for the simulation of a cyclic indirect tensile test. The volume difference for more than 95% aggregates was less than 5% for the two specimens. The difference of stress and strain curves of original and simplified models of the two specimens ranged from 2% to 4%, which indicates that the simulation solutions of the simplified and original microstructures were very close. Furthermore, computational time was significantly reduced due to the simplification of the aggregates. This method facilitates the reliable and highly efficient simulation of asphalt mixtures using the FE method.

Bending analysis of functionally graded rectangular plates using the dual mesh control domain method

The dual mesh control domain method (DMCDM) introduced by Reddy employs one mesh for the approximation of the primary variables (primal mesh) and another mesh for the satisfaction of the governing equations (dual mesh). The present study deals with the extension and application of the DMCDM to functionally graded rectangular plates. The formulation makes use of the traditional finite element interpolation of the primary variables with a primal mesh and a dual mesh to satisfy the integral form of the governing differential equations, the basic premise of the finite volume method. The method is used to analyze bending of through-thickness functionally graded rectangular plates using the first-order shear deformation plate theory. Numerical results are presented to illustrate the methodology, and a comparison of the displacements and stresses computed using the DMCDM with those of the finite element model shows excellent agreement while requiring less formulative (and possibly computational) effort. The influence of the extensional-bending coupling stiffness (due to the through-thickness grading of the material) on the deflections is also brought out.

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling