Low-order Elements Research Articles

A stable implicit nodal integration-based particle finite element method (N-PFEM) for modelling saturated soil dynamics

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

Кинематограф в одном кадре

The article examines the status of the screenshot, a separate static frame, a fragment of the cinematic stream, its relationship to the “whole” of the film, its influence on the spectator’s perception, as well as everyday user practices involving the observation and creation of screenshots. The starting point of the text is an attempt to reconsider the common notion of the screenshot/one-frame as a particle in a subordinate position to the film (and more broadly to the history of cinema). On the basis of texts by Pascal Bonitzer, Sergei Eisenstein, and Roland Barthes, the basic thesis is formulated — the mutual relation between the frame and the film can be constructed differently than the logic of “the frame is a lower-order element nested within the whole film;” on the contrary, the frame and the film in some cases meet as equivalent phenomena or they are superimposed on each other according to the principle of palimpsest. The article deals with three common forms of using frames and screenshots advertising and illustrative shots, the use of film images in sticker packs for messengers, and albums of images in social networks. The question is raised about the communicative plasticity of separate images, the metamorphoses that occur with them when they are taken out of their original context, and their insertion into new artistic and research practices. As such practices, the text pays special attention to two cases — the screenshot project of French director Frank Beauvais and contemporary audiovisual film criticism (based on the works of Johannes Binotto, Carrie Griffith, Daniel McIlwright).

Elastic stability of a lattice of cross-braced shear deformable beams

A plane metamaterial made up of a periodic square grid of shear deformable beams has been braced by crossing cables. This coupling, widely adopted in structural engineering, is here applied with the aim of enhancing the stability performance of a periodic material. Under suitable assumptions, analytical solutions for a rich variety of critical modes and the stability domain are determined by the Bloch wave approach and the effectiveness of our analytical results has been verified by numerical simulations. Furthermore, the experimental response of a specimen, in which hierarchical beams composed by lower order elements are modeled as equivalent shear deformable beams, has shown a satisfactory agreement with our analytical solutions. We argue that our analytical model can be efficiently used to tailor the buckling response of various new metamaterials under general plane stress states.

A general class of enriched methods for the simplicial linear finite elements

Low-order elements are widely used and preferred for finite element analysis, specifically the three-node triangular and four-node tetrahedral elements, both based on linear polynomials in barycentric coordinates. They are known, however, to under-perform when nearly incompressible materials are involved. The problem may be circumvented by the use of higher degree polynomial elements, but their application become both more complex an computationally expensive. For this reason, non-polynomial enriched finite element methods have been proposed for solving engineering problems. In line with previous researches, the main contribution of this paper is to present a general strategy for enriching the standard simplicial linear finite element by non-polynomial functions. A key role is played by a characterization result, given in terms of the non-vanishing of a certain determinant, which provides necessary and sufficient conditions, on the enrichment functions and functionals, that guarantee the existence of families of such enriched elements. We show that the enriched basis functions admit a closed form representation in terms of enrichment functions and functionals. Finally, we provide concrete examples of admissible enrichment functions and perform some numerical tests.

The enriched quadrilateral overlapping finite elements for time-harmonic acoustics

The pronounced numerical dispersions and numerical anisotropy make solutions from the finite element model using low-order elements unreliable for the time-harmonic acoustic problems with fairly large wavenumber. In this work we propose a novel enriched quadrilateral overlapping elements for Helmholtz problems. In this scheme, the original overlapping elements are strengthened by the harmonic trigonometric functions stemming from the spectral techniques. Since all additional degrees of freedom are aligned on the vertex node of every overlapping element, the proposed method can be directly applied to the original finite element model without changing the mesh topology. Because of the enriched approximation space, the proposed method can significantly suppress the numerical dispersions with practically negligible numerical anisotropy, and can be more computationally efficient in providing comparable solution accuracy compared to the original scheme and the classic finite element method. Besides, the linear dependence issue is completely avoided and these enriched overlapping elements are distortion-insensitive. In this work, the original variational formulation is perturbed using the penalty method to impose the essential boundary conditions. Numerical experiments show that the developed method can reduce user interventions in mesh creation and adjustment, and is promising in practical engineering applications for time-harmonic acoustics.

A novel edge center-based gradient-smoothing element method for 2D and 3D coupled thermoelasticity analyses

In this paper, an edge center-based gradient-smoothing element method is proposed for coupled thermoelasticity problems. The main achievement in our work is to construct the linear gradient field inside the element and ensure the consistency of gradient at the junction center of the element, including edge center of triangular element and face center of tetrahedral element. Hence, smoother stress and temperature gradient fields using low-order element are obtained in coupled thermoelasticity analysis without additional post-processing. With the proposed methods, superior accuracy and convergence characteristics are demonstrated compared with other methods through a series of 2D and 3D numerical tests.

A mixed selective edge-based smoothed PFEM with second-order cone programming for geotechnical large deformation analysis

Traditional particle finite element method (PFEM) with a six-node triangular (T6) element using an iteration algorithm often suffers from high computational costs and non-convergence problems. This paper proposes a novel dynamic implicit optimisation based smoothed PFEM for solving geotechnical large deformation problems. The governing equations are treated as an equivalent min–max optimisation that is recast as a second-order cone programming (SOCP) problem. Owing to the particularity of the min–max optimisation problem, which needs to simultaneously interpolate displacement and stress field, a novel mixed constant-stress selective edge-based smoothed strain element is proposed in which strain and stress are constant in the smoothing domain. The two-level mesh repartitioning scheme is implemented into the proposed method to overcome volumetric locking. Next, a mass lumping scheme using nodal integration is developed to improve computational efficiency. An efficient and simple variable mapping scheme is finally proposed and implemented into PFEM to tackle large deformation problems. Several classical statics and dynamics examples are used to validate the proposed method. All results demonstrate that the proposed method only using the lower-order element offers high accuracy, high efficiency, mesh insensitivity and is free of volumetric locking for geotechnical problems under both small and large deformation.

A Bridge between Euclid and Buchberger: (An Attempt to Enhance Gröbner Basis Algorithm by PRSs and GCDs)

This article surveys a very new method of enhancing Buchberger's Gröbner basis algorithm by the PRSs (polynomial remainder sequences) and the GCDs of multivariate polynomials. Let F = { F 1 ,..., F m +1 } ⊂ K[ x , u ] be a given system, where ( x ) = ( x 1 ,..., x m ) and ( u ) = ( u 1 ,..., u n ). Currently, we treat only such F s that are "healthy" (see the text). Let [EQUATION], where [EQUATION], be the reduced Gröbner basis of ideal ( F ) w.r.t. the lexicographic order, to be abbreviated to GB( F ). Let [EQUATION], be such that [EQUATION] is a small multiple of G 1 , and the leading monomial of [EQUATION], is a multiple (hopefully small) of the leading monomial of G i . Our method computes [EQUATION] first, then computes [EQUATION]. Finally, we will apply Buchberger's method to system [EQUATION]. Four new theorems are given. The first and second ones are to compute the lowest-order element of the ideal generated by relatively prime G , H ∈ K[ x , u ], without computing any Spolynomial. The third theorem says that if F is healthy then [EQUATION]. We compute resultants in K[ u ], of F through different routes. Then, by Theorem 3, the resultants will be different multiples of G 1 . Hence, the GCD of them will be a small multiple of G 1 . In the elimination of x through different routes, we obtain sets of similar remainders such that the elements of each set have the same leading variable and nearly the same degrees. We call the leading coefficients of mutually similar remainders an "LCsystem". We eliminate the leading variables of suitably chosen LCsystems. The fourth theorem constructs a polynomial [EQUATION], such that the leading coefficient of [EQUATION] is the GCD of resultants of elements of an LCsystem chosen.

A simple nonconforming tetrahedral element for the Stokes equations

In this paper we apply a nonconforming rotated bilinear tetrahedral element to the Stokes problem in R3. We show that the element is stable in combination with a piecewise linear, continuous, approximation of the pressure. This gives an approximation similar to the well known continuous P2–P1 Taylor–Hood element, but with fewer degrees of freedom. The element is a stable non-conforming low order element which fulfils Korn’s inequality, leading to stability also in the case where the Stokes equations are written on stress form for use in the case of free surface flow.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: High-order flux limiting

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L2-stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes.

A mixed constant-stress smoothed-strain element with a cubic bubble function for elastoplastic analysis using second-order cone programming

This paper proposes a novel optimization-based finite element method with a mixed constant-stress smoothed-strain element to solve elastoplastic problems. The proposed computational framework has three advantages: (1) in the proposed mixed constant-stress smoothed-strain element, the strain is smoothed on the edge-based smoothing domain, and the stress is assumed to be constant in each smoothing domain; (2) the cubic bubble shape function is introduced at the centroid of the element to enrich the displacement field. This makes the proposed approach accurate and free of volumetric locking for low-order elements; (3) the discrete elastoplastic boundary-value problem is formulated as a standard second-order cone problem based on the generalized Hellinger–Reissner variational principle, which is quickly and efficiently solved by the primal-dual interior-point algorithm. The proposed approach is validated on several classical problems of geotechnical engineering: the bearing capacity of footing and pipe under small deformation and slope stability. All the obtained results demonstrate that the proposed method adopting the novel mixed constant-stress smoothed-strain element offers competitively high computational accuracy and efficiency.

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.

Stochastic modeling of the polygonal microstructures of alloys using representative microscopic images

This paper proposes an algorithm to generate a statistically equivalent microstructure of alloys for polygonal mesh-based finite element modeling. Morphological and crystallographic characteristics obtained from scanning electron microscopy images of 2D surface slices are employed to generate the mesh. Titanium alloy microstructures are considered as an example to demonstrate the proposed method. The mean and standard deviation of the grain area distribution obtained from the images are employed as the statistical parameters. The algorithm uses the image data and applies perturbation using a distortion factor and Voronoi tessellation to generate the statistically equivalent polygonal microstructure that captures the randomness of grain shape and size iteratively. The grain area distribution of the simulated microstructure closely matches with the actual distribution. Lower-order elements (quadrilateral) on the domain boundary get introduced that may create artifacts when analyzed with the finite element method. A method to eliminate these artifacts is presented with an analysis of stress concentrations occurring due to grain orientation mismatch. Effective mechanical properties, deformation, and fracture behavior as a function of various microstructural parameters like grain area and crystallographic orientations can be analyzed using the polygonal finite element mesh.

Analysis of two-dimensional acoustic radiation problems using the finite element with cover functions

Due to the pollution errors in acoustic analysis, the requirement of very refined meshes at high wavenumbers limits the practical application of the conventional finite element method (FEM). In this work, the three-node linear triangular element with linear interpolation cover functions (FEwC) is applied to solve acoustic radiation problems in two dimensions. The present FEwC has considerable generality since the polynomial interpolation cover functions can be easily implemented without changing the mesh topology. As the original linear approximation space is enriched by the cover functions, the FEwC can significantly alleviate the pollution errors and increase the convergence rate of the original low-order element in acoustic computation. We show the proper imposition of the Dirichlet boundary conditions in acoustic problems and illustrate an effective iterative algorithm to resolve the system equations with a singular coefficient matrix. The Dirichlet-to-Neumann mapping technique is used to truncate the infinite problem domain into a bounded computational domain for the exterior problems and the Sommerfeld radiation condition then can be satisfied at infinity. Several numerical experiments demonstrate that the present scheme is a promising numerical tool for acoustic radiation problems.

Effect of numerical integration on a new rotated nonconforming quadrilateral element

A new nonparametric nonconforming quadrilateral finite element is used to approximate the general second-order elliptic problem in two dimensions. Some optimal numerical integration formulas are presented and analyzed. These formulas are derived on a reference quadrilateral which can be linearly mapped to a physical quadrilateral. The novelty of these formulas is that they only involve two quadrature nodes which excludes even a Q1 unisolvent set, and they are not required to be exact for all the shape functions. Numerical tests show that the presented quadrature rules can also be used coupled with other low-order elements.

Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

Improving convergence by optimizing the condition number of the stiffness matrices arising from least-squares finite element methods

The least-squares finite element method (LSFEM) has drawn much attention with desirable properties, such as always symmetric positive-definite stiffness matrix and approximation of primal and non-primal variables with equivalent accuracy. Despite a condition number comparable to that of Galerkin FEM, of O(h−2), it is sometimes found that LSFEM with low-order elements does not show a satisfactory accuracy of solution. In this report, we propose an easy-to-implement technique that optimizes the condition number of the stiffness matrix. The least-squares functional is applied to the first-order governing equations and a linear combination of the weak form domain and boundary equations, with normalization (i.e., weighting) parameters is constructed. The impact of these parameters on the condition number of the resulting global stiffness matrix is proved to be independent of mesh size or shape function order. By use of a global optimization scheme that tunes these parameters, we observe notable decrease in the condition number compared to cases where the least-squares principle is directly applied to the governing equations, and hence convergence is significantly improved. It is further shown that the technique can be applied to the elemental stiffness matrix for a computationally efficient means of determining normalization parameters. To show the general applicability of the method, we also apply this technique to solve for a two-dimensional (2D) lid-driven flow problem and to calculate the electromagnetic field distribution in a 2D rectangular resonant cavity. In both problems, the LSFEM solution accuracy is poor prior to optimization of weighting parameters, and accuracy does not increase much by mesh refinement alone. Moreover, the process can be generalized to Galerkin FEM and other methods to improve convergence properties.

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III

A weak Galerkin (WG) finite element method without stabilizers was introduced in Ye and Zhang (2020) on polytopal mesh. Then it was improved in Ye and Zhang (2021) with order one superconvergence. The goal of this paper is to develop a new stabilizer free WG method on polytopal mesh. This method has convergence rates two orders higher than the optimal convergence rates for the corresponding WG solution in both an energy norm and the L2 norm. The numerical examples are tested for low and high order elements in two and three dimensional spaces.

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

AbstractA new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity‐pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal‐order error estimates are established for the corresponding numerical approximation in various norms. The numerical examples are tested for low and high order elements up to the degree four in 2D and 3D spaces.

Nonconforming finite element method for coupled Poisson–Nernst–Planck equations

AbstractA nonconforming finite element method (FEM) is developed and investigated for the coupled Poisson–Nernst–Planck (PNP) equations with low order element. Then, by use of the special properties of this element, that is, the interpolation operator is equivalent to its projection operator, and the consistency error estimate can reach order of O(h2) which is one order higher than that of its interpolation error estimates when the exact solution belongs to H3(Ω), the superclose estimates of order O(h2) and O(h2 + τ) in the broken H1‐norm are derived with new techniques for the semidiscrete scheme and backward Euler fully discrete scheme, respectively. Further, through employing interpolation postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some numerical results are provided to confirm the theoretical analysis. It seems that our results have never been found in the existing literature. Here h and τ denote the mesh size and time step, respectively.