An improved algorithm for developing ‘special’ finite element shape functions

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.

This paper presents a finite element-least square point interpolation method (FE-LSPIM) for solving 2D acoustic problem by synthesizing the attractive features of finite element and least-square point interpolation shape functions using the concepts of partition of unity (PU) methods. In the present method, the acoustic domain is discretized using quadrilateral element, and the shape functions of the quadrilateral element are used for PU and the least-square point interpolation method (LSPIM) for local approximation. This enables the proposed method to inherit the compatibility properties of finite element method and the quadratic polynomial completeness properties of meshfree methods, so that the finite element-least-square point interpolation method (FE-LSPIM) will greatly reduce the numerical dispersion error because the numerical dispersion error is essentially caused by the �overly-stiff� nature of the FEM model. Numerical results for benchmark problems show that, the FE-LSPIM achieves more accurate results and higher convergence rates as compared with the corresponding finite elements and Element-free Galerkin method (EFGM), especially for high wave number and irregular meshes

A new approach to optimize variable axial composite shells: A metaheuristic procedure using the finite element method

Micro-mass sensors have attracted increasing attention in the field of biomolecular and chemical detection. It has been found that the size, shape, and geometry of the structure may affect the performance of the sensor. As a result, a topology optimization methodology is proposed in this paper for the design of micro-mass sensors. A phase-field function controlled by nodal variables and finite element shape functions is used to describe the configuration of a sensor in the constructed optimization problem. The design goal is to maximize the mass detection sensitivity. On the basis of these formulations, an optimization algorithm is constructed using the finite element method and the method of moving asymptotes. Numerical examples are presented to demonstrate the validity of the proposed problem formulation. The results suggest that the performance of the micro-mass sensor can be improved by using the proposed approach.

We 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.

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.

Simulation of two dimensional unilateral contact using a coupled FE/EFG method

High-fidelity fluid-structure interaction applied to static aeroelasticity in transonic flows

A note on the use of first order quadrilateral elements in axisymmetric analysis

Mixed finite elements based on superconvergent patch recovery for strain gradient theory

This manuscript presents a simplified approach to adhesive joints calculation. Aviation and space rocket engineering constructions joints deals with coaxial cylindrical pipes that can be connected by adhesive with advantages in terms of tightness, aerodynamic efficiency, manufacturability, low weight. Glue joints are, therefore, considered and simplified in this article by the analytical calculation of the shape functions for a macro-element applied to the Volkersen’s theory. The mathematical solution makes possible to calculate the explicit form of the stiffness matrix of the macro-element. The analysis proposed shows the perfect matching of the solution for a single element under classical mechanical loads.

This paper firstly presents numerical analyses of functionally graded porous plates/shells with graphene platelets (GPLs) reinforcement using a novel four-node quadrilateral element with five degrees of freedom per node, namely SQ4P, based on the first-order shear deformation theory and Chebyshev polynomials. The novelty of the present element is to use the high-order shape functions which satisfy the interpolation condition at the points based on Chebyshev polynomials to build the new flat four-node element for analysis of plate/shell structures. The Chebyshev polynomials are a sequence of orthogonal polynomials that are described recursively and the values of these polynomials belong to the interval [−1,1] as well as vanish at the Gauss points. Full Gauss quadrature rule is used to establish the stiffness matrix, geometric stiffness matrix, mass matrix and load vector. Various dispersions of GPLs and internal pores into the metal matrix through the thickness of structure are considered with the rule of a mixture and the Halpin–Tsai model for evaluating effective material properties across the thickness. The influence of weight fraction, porosity coefficient and dimensions of GPLs, distribution of GPLs and porosity into metal matrix are fully studied via several numerical examples from static bending to free vibration and buckling response. Numerical results and comparison with other solutions from available references suggest that the present element has enough reliability and validity to use in structural analysis. With regular and irregular meshes, these results are in close agreement with the exact solutions by using the suitable value for the order of the shape functions.

For computational efficiency, partition of unity enrichments are preferably localized to the sub‐domains where they are needed. It is shown that an appropriate construction of the elements in the blending area, the region where the enriched elements blend to unenriched elements, is often crucial for good performance of local partition of unity enrichments. An enhanced strain formulation is developed which leads to good performance; the optimal rate of convergence is achieved. For polynomial enrichments, it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence. The methods are illustrated by several examples. The examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element, but other enrichments are also considered. Results show that both methods provide optimal rates of convergence. Copyright © 2003 John Wiley & Sons, Ltd.

ABSTRACTTo address the substantial computational cost in modeling the mechanical behavior of large‐scale architected lattice structures, this work introduces a concurrent multiscale framework: the generalized summation rule‐based nonlocal quasicontinuum (GSR‐QC) method. The core innovation is a generalized summation rule that enables accurate coarse‐graining consistent with general finite element shape functions. While a few existing approaches have explored summation rules for higher‐order interpolation, such efforts remain limited and are typically tailored to specific element types or applications. In contrast, the proposed framework provides a unified and systematic approach that ensures compatibility with general shape functions, significantly enhancing the flexibility and applicability of nonlocal quasicontinuum methods. The GSR‐QC method features: (1) constitutive‐model consistency, employing the same discrete lattice model in both fully resolved and coarse‐grained regions; (2) shape‐function‐consistent energy sampling, aligned rigorously with the interpolation order of general finite elements; and (3) interfacial compatibility, enabling seamless energy and force transfer across regions of differing resolutions without additional interface treatment. The performance of GSR‐QC is validated using bilinear quadrilateral and quadratic triangular elements across benchmark problems—including uniaxial tension, clamped bending, three‐point bending, and crack propagation in truss‐based lattice structures—demonstrating good accuracy. Additionally, the error analysis and convergence behavior of GSR‐QC are investigated.

In this paper, a computational approach is adopted to solve the optimal control and optimal parameter selection problems of a rotating flexible beam fully covered with active constrained layer damping (ACLD) treatment. The beam rotates in a vertical plane under the gravitational effect with variable angular velocity and carries an end mass. Tangent coordinate system and the moving coordinate system are used in the system modeling. Due to the highly nonlinear and coupled characteristics of the system, a relative description method is used to represent the motion of the beam and the motion equations are set up by using relative motion variables. Finite element shape functions of a cantilever beam [1] are used as the displacement shape functions in this study. Lagrangian formulation and Raleigh-Ritz approach [2] are employed to derive the governing equations of motion of the nonlinear time-varying system. The problem is posed as a continuous-time optimal control problem. The control function parameters are the control gains. The two system parameters are the thickness of the constraining layer and the viscoelastic material layer. The software package MISER3.2, which is based on the Control Parametrization and the Control Parametrization Enhancing Transform (CPET) techniques is used to solve the combined problems. The optimal solution takes the end deflection, control voltage and the total weight into account. Results show that substantial improvements are obtained with ACLD as compared to the passive constrained layer damping (PCLD) treatment.