Quadratic Shape Functions Research Articles

Coupled field modeling of thermoresponsive hydrogels with upper/lower critical solution temperature

An inf–sup stable FE formulation for the thermo-chemo-mechanical simulation of thermoresponsive hydrogels is herein proposed by approximating the displacement field via quadratic shape functions and both the chemical potential (fluid pressure) and the temperature fields by linear functions. The formulation is implemented into a stable thermo-chemo-mechanical user-element subroutine (UEL) in Abaqus, denoted as Q2Q1Q1. The proposed formulation has been validated in relation to thermoresponsive hydrogels to interpret several examples of transient diffusion-driven swelling deformations. First, the upper/lower critical solution temperature behaviors of thermoresponsive hydrogels has been captured, studying several peculiarities comprising the diffusion length influence at the instantaneous loading state and the overlooked influence of the mass flux and the hyperelastic stretching on the temperature field. Subsequently, numerical analysis have been conducted in order to investigate the impact of temperature-dependent swelling ratio on the mechanical behavior of spheres undergoing compression. The accuracy of the proposed formulation has been assessed by numerically replicating the seminal experiments that explore the influence of crosslinking density on the thermally driven swelling of PNIPAAm hydrogels.

Analysis and simulation of the (2 + 1)-dimensional Fisher’s reaction-diffusion equation with higher order finite element method

In this article, a finite element analysis of Fisher’s reaction-diffusion equation is performed, by using quadratic shape elements. The higher-order elements in the finite element method (FEM) are known to yield better results in the approximation of solutions. However, such elements are less explored in the literature. Therefore, quadratic shape functions are taken into consideration for the finite element analysis of Fisher’s reaction-diffusion equation. A priori error estimates for the semi-discrete solution are derived by applying the Galerkin approximation. For the fully discrete solution, a priori error estimates are derived using the Crank-Nicolson method and the predictor-corrector scheme. Also, the stability of the time-discrete scheme is analyzed by the energy technique. Third-order convergence for space discretization in L 2 ( Ω ) norm and second-order convergence in H 1 ( Ω ) norm are observed. Furthermore, second-order convergence for time discretization is also found. Finally, we validate our theoretical results by considering a variety of numerical examples in one and two-dimensional spaces, which also illustrate the reliability of the developed finite element algorithm. Through comparison tables, 3D plots, and convergence plots, the results of the current study are compared with the results from the existing literature.

Higher order Galerkin finite element method for [formula omitted]-dimensional generalized Benjamin–Bona–Mahony–Burgers equation: A numerical investigation

In this article, we study solitary wave solutions of the generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation using higher-order shape elements in the Galerkin finite element method (FEM). Higher-order elements in FEMs are known to produce better results in solution approximations; however, these elements have received fewer studies in the literature. As a result, for the finite element analysis of the gBBMB equation, we consider Lagrange quadratic shape functions. We employ the Galerkin finite element approximation to derive a priori error estimates for semi-discrete solutions. For fully discrete solutions, we adopt the Crank–Nicolson approach, and to handle nonlinearity, we utilize a predictor–corrector scheme with Crank–Nicolson extrapolation. Additionally, we perform a stability analysis for time using the energy method. In the space, O(h3) convergence in L2(Ω) norm and O(h2) convergence in H1(Ω) norm are observed. Furthermore, an optimal O(Δt2) convergence in the maximum norm for the temporal direction is also obtained. We test the theoretical results on a few numerical examples in one- and two-dimensional spaces, including the dispersion of a single solitary wave and the interaction of double and triple solitary waves. To demonstrate the efficiency and effectiveness of the present scheme, we compute L2 and L∞ normed errors, along with the mass, momentum, and energy invariants. The obtained results are compared with the existing literature findings both numerically and graphically. We find quadratic shape functions improve accuracy in mass, momentum, energy invariants and also give rise to a higher order of convergence for Galerkin approximations for the considered nonlinear problems.

Improved dynamic analysis of shear deformable shells using the hybrid displacement boundary element method

In this paper, the dynamic analysis of shear deformable shells was conducted using the hybrid displacement boundary element method (HD-BEM). The approach involved utilizing the fundamental solution as a trial function to derive the 8-node element stiffness matrix, while approximating the boundary variables with quadratic shape functions. To ensure efficient computations and avoid singular integrals, the source points were strategically positioned outside the domain. For accurate calculations of the proposed 8-node element mass matrix, only four Gauss points were relied upon. It is important to note that both the element mass and stiffness matrices demonstrate symmetry. Throughout the study, free, forced, and harmonic analyses were considered. Encouragingly, the results obtained from the study exhibited exceptional accuracy levels, even when employing a coarse discretization approach.

Lateral Response of 2 x 2 Pile Group Embedded in Cohesive Soil Near Slope

This work studies the response of a 2 x 2 pile group near a slope subjected to lateral load. Three-dimensional, non-linear finite element analysis is carried out. The entire code is developed in FORTRAN 90. Bending-dominated pile and pile caps are represented by 20 node elements with quadratic shape function, and shear-dominated soil elements are represented by eight-node elements with linear shape function. Sixteen node interface element of zero thickness is used for stress transfer between pile and soil. The outcome highlights the influence of slope parameters like slope angle and edge distance on the response of the 2 x 2 pile group near a slope. The lateral displacement and maximum bending moments are increased with an increase in slope. However, its effect reduces with an increase in slope distance. The point of zero-shear along the pile's depth shifts deeper with an increase in slope.

An efficient Artificial Neural Network algorithm for solving boundary integral equations in elasticity

This study presents a novel numerical integration technique based on Artificial Neural Network (ANN) algorithms to overcome intrinsic limitations characterizing the Boundary Element Method (BEM). The proposed approach, taking advantage of some peculiar properties of the BEM equations, provides an effective alternative to traditional numerical techniques for evaluating the integrated kernels required to compute the displacements and stresses of a two-dimensional solid. Assuming isotropy and homogeneity, and modeling both the geometry and the mechanical parameters using quadratic shape functions, all the integrals in the classical BEM formulation can be expressed as the sum of two terms that are independent of the constitutive properties and solely dependent on four geometric parameters: three components of two distance vectors and a parameter representing the element’s curvature. This interesting property of boundary integral equations in elasticity makes them particularly amenable to numerical evaluation using artificial neural networks. Results from numerical tests, which were conducted using increasingly complex integrals, demonstrate the high precision of the proposed approach as long as the integration and collocation points are sufficiently separated to avoid issues with singularity.

Two-Dimensional C-V Heat Conduction Investigation of an FG-Finite Axisymmetric Hollow Cylinder

In the present work, we implement a graded finite element analysis to solve the axisymmetric 2D hyperbolic heat conduction equation in a finite hollow cylinder made of functionally graded materials using quadratic Lagrangian shape functions. The graded FE method is verified, and the simple rule of the mixture with power-law volume fraction is found to enhance the effective thermal properties’ gradation along the radial direction, including the thermal relaxation time. The effects of the Vernotte numbers and material distributions on temperature waves are investigated in depth, and the results are discussed for Fourier and non-Fourier heat conductions, and homogeneous and inhomogeneous material distributions. The homogeneous cylinder wall made of SUS304 shows faster temperature wave velocity in comparison to the ceramic-rich cylinder wall, which demonstrates the slowest one. Furthermore, the temperature profiles along the radial direction when n = 2 and n = 5 are almost the same in all Ve numbers, and by increasing the Ve numbers, the temperature waves move slower in all the material distributions. Finally, by tuning the material distribution which affects the thermal relaxation time, the desirable results for temperature distribution can be achieved.

Bending, buckling and free vibration analyses of shallow-to-deep FG curved sandwich beams using a global–local refined shear deformation theory

This paper investigates the bending, vibration, and buckling behaviour of functionally graded (FG) curved sandwich beams using a global–local refined shear deformation theory. Material properties of these beams are varied through the thickness according to the power-law distribution. A beam element with a combination of Hermite cubic and quadratic Lagrange shape functions is developed to solve the problems. The obtained results are compared with the results from finite element software (ABAQUS), and other higher-order beam theories. It can be observed that the present model can predict accurately static, vibration, and buckling responses of both shallow and deep FG sandwich beams with arbitrary boundary conditions.

Finite-element method based solver for an evaporating sessile droplet on a heated surface

We report the development of a finite-element-based solver to compute transport of mass, momentum and energy during evaporation of a sessile droplet on a heated surface. The evaporation is assumed to be quasi-steady and diffusion-limited. The heat transfer between the droplet and substrate and mass transfer of liquid-vapor are solved using a two-way coupling. In particular, here, we develop and implement the formulation of fluid flow inside the droplet in the model. The continuity and Navier–Stokes equations are solved in axisymmetric, cylindrical coordinates. Jump velocity boundary condition is applied on the liquid–gas interface using the evaporation mass flux. The governing equations are discretized in the framework of the Galerkin weight residual approach. A mesh of finite triangular elements with six nodes is utilized, and quadratic shape functions are used to obtain the second-order accurate numerical solution. Two formulations, namely, penalty function and velocity pressure, are employed to obtain discretized equations. The numerical results are the same using both methods, and the latter is around 30–50% faster than the former for the cases of refined grid. Computed flow fields are in excellent agreement with published results. The solver’s capability is demonstrated by solving the internal flow field for a case of a heated substrate.

Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids

Flexoelectricity is an electromechanical coupling between the electric field and the mechanical strain gradient, as well as between the mechanical strains and the electric field gradient, observed in all dielectric materials, including those with centrosymmetry. Flexoelectricity demands C1-continuity for straightforward numerical implementation as the governing equations in the gradient theory are fourth-order partial differential equations. In this work, an alternative collocation-based mixed finite element method for direct flexoelectricity is used, for which a newly developed quadratic element with a high capability of capturing gradients is introduced. In the collocation method, mechanical strains and electric field through independently assumed polynomials are collocated with the mechanical strains and electric field derived from the mechanical displacements and electric potential at collocation points inside a finite element. The mechanical strain gradient and electric field are obtained by taking the directional derivative of the independent mechanical strain and electric field gradients. However, an earlier proposed linear element is unable to capture all mechanical strain gradient components and, thus, simulate flexoelectricity correctly. This problem is solved in the present work by using quadratic shape functions for the mechanical displacements and electric potential with fewer degrees of freedom than the traditional mixed finite element method. A Fortran user-element code is developed by the authors: first, for the linear and, after that, for the quadratic element. After verifying the linear element with numerical results from the literature, both linear and quadratic elements’ behaviors are tested for different problems. It is shown that the proposed second-order collocation-based mixed FEM can capture the flexoelectric behavior better compared to the existing linear formulations.

Efficient fundamental solution based finite element for 2-d dynamics

In this paper, an efficient finite element is developed for the dynamic analysis of 2-D elasticity problems. Unlike the conventional direct or indirect formulations, the proposed integral equation is based on minimizing the relevant energy functional. In doing so, variational methods are used. The proposed element stiffness matrix is obtained by a modified hybrid displacement variational statement, taking the fundamental solution as a trial function. Quadratic shape functions are used for the approximation of the boundary variables. To avoid singularities, the source points are located outside the problem domain. The element mass matrix is computed using the relevant element shape functions. Only four Gauss points are needed for accurate computation of the domain integral related to the mass matrix computation. The proposed element is applicable for: free vibration, forced vibration, and harmonic analysis as demonstrated by the presented numerical examples. The obtained results are very promising, and the accuracy level is excellent.

Penalty C0 8-node quadrilateral and 20-node hexahedral elements for consistent couple stress elasticity based on the unsymmetric finite element method

In this paper, the penalty unsymmetric finite element framework for the consistent couple stress theory is derived from the virtual work principle. The C1 continuity requirement is satisfied in weak form by using the penalty function method to constrain the independently introduced rotations for approximating the mechanical rotations, enabling the utilization of C° continuous interpolations for designing the element displacement without the loss of convergence property. Within the proposed framework, 8-node quadrilateral element and 20-node hexahedral solid element are constructed for analyzing the size-dependent mechanical responses of consistent couple stress elasticity materials. In these developments, the quadratic serendipity isoparametric shape functions are enriched by the rotation degrees of freedom for determining the test functions, whilst the metric stress functions that are derived from the concerned equilibrium equations are used to design the trial functions. A series of numerical benchmarks are examined for verifying their effectiveness and accuracy. It is shown that the elements can efficiently capture the size dependences, exhibiting good accuracy and low susceptibility to mesh distortion.

A new higher-order mixed four-node quadrilateral finite element for static bending analysis of functionally graded plates

In this paper, a new C0 four-node quadrilateral plate element (MiQ4) is proposed for the static bending analysis of functionally graded (FG) plates based on a C0 higher-order shear deformation theory. By introducing a quadratic shape function for higher-order terms and a constant shear result, which is eliminated at the element level, the transverse shear stresses are free on two surfaces of the plates. The present finite element model has high accurateness and a fast rate of convergence, especially for both thin and very thin plates. In addition, the proposed element is insensitive to distorted mesh, hence, it is compatible with the investigation of complex geometrical plates. Although the proposed element does not need any shear correction coefficients and selected or reduced integrations, the shear-locking phenomenon does not appear even for very thin plates. Therefore, the computation cost and time are reduced and the spurious deformation or hourglass modes are avoided. A large number of examples have been exhibited that help to validate the convergence rate, correctness, and reliability of the proposed element. Then the present element is applied for analysis of FG structures to demonstrate the potential applications of the present plate element. Significantly, the paper provides a good element that can be used in many areas of engineering for the design, testing, analysis, and optimization of complex structures to reduce the computation cost and time as well as increase the reliability of the numerical results.

Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling

Numerical simulation and inversion imaging are essential in geophysics exploration. Fourier transform plays a vital role in geophysical numerical simulation and inversion imaging, especially in solving partial differential equations. This paper proposes an arbitrary sampling Fourier transform algorithm (AS-FT) based on quadratic interpolation of shape function. Its core idea is to discretize the Fourier transform integral into the sum of finite element integrals. The quadratic shape function represents the function change in each element, and then all element integrals are calculated and accumulated. In this way, the semi-analytical solution of the Fourier oscillation operator in each integral interval can be obtained, and the Fourier transform coefficient can be calculated in advance, so the algorithm has high calculation accuracy and efficiency. Based on the one-dimensional (1D) transform, the two-dimensional (2D) transform is realized by integrating the 1D Fourier transform twice, and the three-dimensional (3D) transform is realized by integrating the 1D Fourier transform three times. The algorithm can sample flexibly according to the distribution of integrated values. The correctness and efficiency of the algorithm are verified by Fourier transform pairs. The AS-FT algorithm is applied to the numerical simulation of magnetic anomalies. The accuracy and efficiency are compared with the standard Fast Fourier transform (standard-FFT) and Gauss Fast Fourier transform (Gauss-FFT). It shows that the AS-FT algorithm has no edge effects and has a higher computational speed. The AS-FT algorithm has good adaptability to continuous medium, weak magnetic catastrophe medium, and strong magnetic catastrophe medium. It can achieve the same as or even higher accuracy than Gauss-FFT through fewer sampling points. The AS-FT algorithm provides a new means for partial differential equation solution in geophysics. It successfully solves the boundary problems, which makes it an efficient and high-precision Fourier transform approach with promising applications. Therefore, the AS-FT algorithm has excellent advantages in solving partial differential equations, providing a new means for solving geophysical forward and inverse problems.

Active auto-adaptive metamaterial plates for flexural wave control

In this paper, a novel design concept for active auto-adaptive metamaterial (AAAMM) plates is proposed based on an active auto-adaptive (AAA) control strategy guided by the particle swarm optimization (PSO) technique. The AAAMM plates consist of an elastic base plate and two periodic arrays of piezoelectric patches. The periodic piezoelectric patches placed on the bottom plate surface act as sensors, while the other ones attached on the top plate surface act as actuators. A two-dimensional (2D) simplified metamaterial (MM) plate model is established by the Hamilton principle. The finite element method (FEM) with quadratic shape functions is adopted for numerically solving the 2D simplified MM plate model, which is validated by two more accurate but also much more complicated FEM models using three-dimensional (3D) piezoelectric and 3D elastic solid elements, and by its 2D counterpart model solved by the plane wave expansion (PWE) method. The conventional displacement, acceleration and velocity feedback control methods are introduced and analyzed. Then, a novel AAA control strategy based on combining the displacement and acceleration feedback control methods and guided by the PSO technique is developed. The main objective of the present paper aims at the vibration or flexural wave suppression and isolation of elastic plate structures based on an auto-adaptive control strategy, which has wide-range engineering applications. Numerical results will be presented and discussed to show that the proposed AAAMM plates can automatically and intelligently evolve different feedback control schemes to adapt to different stimulations on demand. Compared to the conventional MM plates, the proposed AAAMM plates exhibit improved and enhanced band-gap characteristics and suppression performance for flexural waves at frequencies inside and outside the band-gaps.

Variational quadratic shape functions for polygons and polyhedra

Solving partial differential equations (PDEs) on geometric domains is an important component of computer graphics, geometry processing, and many other fields. Typically, the given discrete mesh is the geometric representation and should not be altered for simulation purposes. Hence, accurately solving PDEs on general meshes is a central goal and has been considered for various differential operators over the last years. While it is known that using higher-order basis functions on simplicial meshes can substantially improve accuracy and convergence, extending these benefits to general surface or volume tessellations in an efficient fashion remains an open problem. Our work proposes variationally optimized piecewise quadratic shape functions for polygons and polyhedra, which generalize quadratic P 2 elements, exactly reproduce them on simplices, and inherit their beneficial numerical properties. To mitigate the associated cost of increased computation time, particularly for volumetric meshes, we introduce a custom two-level multigrid solver which significantly improves computational performance.

Quadrilateral scaled boundary spectral shell elements for static and dynamic analyses of functionally graded plates and shells

This paper presents new shell elements for static and dynamic analyses of functionally graded plates and shells with the material properties varying through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. In the formulation, a shell element is treated as a three-dimensional linear elastic body and its middle surface is represented with a quadrilateral spectral element. Along the thickness direction, the shell geometry is described by scaling the middle surface while the displacements are approximated by interpolating the displacements on the top, middle and bottom surfaces using quadratic Lagrange shape functions. The assumed natural strain method is applied to eliminate transverse shear locking, membrane locking and curvature thickness locking. The developed shell elements are featured by requiring only the shell mid-surface to be discretized, containing no rotational degree of freedom, adopting the full three-dimensional constitutive law and involving only the in-plane numerical integration. The validity and performance of the formulation are demonstrated through benchmark examples covering plates, cylindrical shells and spherical shells with varying span-to-thickness ratio and material gradation index. Static solutions of two functionally graded shell problems under gravity loads are also provided.

Quadrilateral scaled boundary spectral shell elements with assumed natural strains

This paper presents new variable-order shell elements that require only the shell mid-surface to be discretized, contain no rotational degree of freedom and adopt the full three-dimensional constitutive relationship. In the formulation, a shell element is treated as a three-dimensional continuum undergoing small deformations and its middle surface is represented by a quadrilateral spectral element. Along the thickness direction, the shell geometry is described by scaling the middle surface and the displacements are expressed approximately by interpolating the displacements on the top, middle and bottom surfaces with quadratic Lagrange shape functions. The assumed natural strain method is applied to eliminate transverse shear locking, membrane locking and curvature thickness locking while the volumetric locking is alleviated by elevating the element order. Numerical examples demonstrate that the developed shell elements are superior in applicability, accuracy and efficiency.

Fast numerical simulation of 2D gravity anomaly based on nonuniform fast Fourier transform in mixed space-wavenumber domain

Wavenumber domain gravity modeling is typically based on the fast Fourier transform (FFT), requiring uniform sampling in both space and wavenumber domains and grid expansion to guarantee the accuracy. In this paper, we develop an efficient and highly accurate 2D forward modeling algorithm based on the nonuniform fast Fourier transform (NUFFT) in the hybrid space and wavenumber domain. To retain the accuracy of space domain methods, the Fourier transform is carried out only in the horizontal direction based on NUFFT and no transform is applied in the vertical direction (make the discretization in the vertical direction flexible). Thus, the original gravity modeling problem is transformed into a sequence of independent space domain 1D integral equations, which can be solved in parallel efficiently. For the 1D integral in the space domain, the density is represented by quadratic shape function and the independent equations can be solved analytically. The application of the NUFFT algorithm allows for the nonuniform discretization of a model and improves calculation efficiency while ensuring calculation accuracy. Based on a 2D prism model, the accuracy of the proposed algorithm is verified and its numerical performance in terms of efficiency and accuracy is compared with the standard FFT and Gauss-FFT methods, indicating that at the same accuracy our algorithm is most efficient. Then, three more complex models and one BP density model are designed to further demonstrate the good numerical performance of the proposed algorithm.

Shape-function-based nonuniform Fourier transforms for seismic modeling with irregular grids

Multidimensional Fourier transform on an irregular grid is a useful tool for various seismic forward problems caused by complex media and wavefield distributions. Using a shape-function-based strategy, we have developed four different algorithms for 1D and 2D nonuniform Fourier transforms, such as two high-accuracy Fourier transforms (the linear shape-function-based Fourier transform [LSF-FT] and the quadratic shape-function-based Fourier transform [QSF-FT]) and two nonuniform fast Fourier transforms (NUFFTs) (the linear shape-function-based NUFFT [LSF-NUFFT] and the quadratic shape-function-based NUFFT [QSF-NUFFT]), respectively, based on their linear and quadratic shape functions. The main advantage of incorporating shape functions into the Fourier transform is that triangular elements can be used to mesh any complex wavefield distribution in the 2D case. Therefore, these algorithms can be used in conjunction with any irregular sampling strategies. The accuracy and efficiency of the four nonuniform Fourier transforms are investigated and compared by applying them in frequency-domain seismic wave modeling. All of the algorithms are compared with the exact solutions. Numerical tests indicate that the quadratic shape-function-based algorithms are more accurate than those based on the linear shape function. Moreover, LSF-FT/QSF-FT exhibits higher accuracy but much slower calculation speed, whereas LSF-NUFFT/QSF-NUFFT is highly efficient but has lower accuracy at near-source points. In contrast, a combination of these algorithms, by using QSF-FT at near-source points and LSF-NUFFT/QSF-NUFFT at others, achieves satisfactory efficiency and high accuracy at all points. Although our tests are restricted to seismic models, these improved NUFFT algorithms may also have potential applications in other geophysical problems, such as forward modeling in complex gravity and magnetic models.