Derivatives Of Shape Functions Research Articles

A linear smoothed quadratic finite element for buckling analysis of laminated composite plates

In this paper, a linear smoothing scheme over eight-node Reissner-Mindlin plate element under the framework of the CS-FEM is employed to buckling analysis of laminated composite plates based on the first-order shear deformation theory. The modified stain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor's expansion. Isoparametric mapping and the computation of interior derivatives of shape function is not required in the proposed method, furthermore, all the computation are based on the global Cartesian coordinates. Some numerical examples are given at the end to demonstrate that the present method has good performance to alleviate the shear-locking phenomenon and improve the quality of the solutions with distorted meshes.

A rotation-free Hellinger-Reissner meshfree thin plate formulation naturally accommodating essential boundary conditions

A rotation-free Hellinger-Reissner meshfree thin plate formulation is proposed to naturally accommodate the essential boundary conditions in a variationally consistent way. In this approach, the Galerkin weak form is established based upon the Hellinger-Reissner variational principle, where the bending moment is expressed as the second order smoothed gradients which inherently embed the integration constraint and fulfill the variational consistency condition. Owing to the Hellinger-Reissner variational principle, the essential boundary conditions naturally arise in the weak form. Accordingly, the enforcement of essential boundary conditions has a similar form as that of the Nitsche's method, i.e., both have standard consistent and stabilized terms. Compared with the Nitsche's method, the costly second and higher order derivatives of traditional meshfree shape functions are replaced by the fast-evaluated second order smoothed gradients and their derivatives. Meanwhile, the stabilized term in proposed method does not involve any artificial parameter and thus eliminates the stabilization parameter-dependent issue in the Nitsche's formulation. Several examples are presented to illustrate the convergence, accuracy and efficiency of the proposed rotation-free Hellinger-Reissner meshfree thin plate formulation.

The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation

In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control theory. We show that for a class of partial differential equations, the shape-dependent state variable can be differentiated with respect to the topology, thus leading to a linearised system resembling those occurring in standard optimal control problems. However, a lot of care has to be taken when handling the regularity of the solutions of this linearised system. In fact, we should expect different notions of (very) weak solutions, depending on whether the main part of the operator or its lower order terms are being perturbed. We also study the relationship with the topological state derivative, usually obtained through classical topological expansions involving boundary layer correctors. A feature of the topological state derivative is that it can either be derived via Stampacchia-type regularity estimates or alternately with classical asymptotic expansions. It should be noted that our approach is flexible enough to cover more than the usual case of point perturbations of the domain. In particular, and in the line of (Delfour in SIAM J Control Optim 60(1):22–47, 2022; J Convex Anal 25(3):957–982, 2018), we deal with more general dilatations of shapes, thereby yielding topological derivatives with respect to curves, surfaces or hypersurfaces. To draw the connection to usual topological derivatives, which are typically expressed with an adjoint equation, we show how usual first-order topological derivatives of shape functionals can be easily computed using the topological state derivative.

A New Rectangular Finite Element for Static and Dynamic Analysis of Arbitrarily Tapered Plates

This paper presents the formulation of a new efficient and conforming rectangular finite element for analysis of thin plates with any arbitrary variation of thickness along both edges. Shape functions of this new element are derived from multiplying shape functions of non-prismatic Euler–Bernoulli beam extracted from basic displacement functions. To provide [Formula: see text] consistency along the edges of elements, twist is added to conventional degrees of freedom, namely deflection and slopes resulting in an element with 16 degrees of freedom. The proposed element is used to solve various static and dynamic problems, and it is seen that the convergence of a new formulation occurs with much fewer elements compared to existing finite elements as a direct result of considering the variation of geometry in the derivation of shape functions, which renders the formulation competitive in both exactness and economy.

Smoothed floating node method for modelling cohesive fracture in quasi-brittle materials

This paper presents a numerical framework for implementation of cohesive zone model with smoothed floating node method (SFNM) for failure analysis of quasi-brittle materials. The nonlinear behavior of material inside the fracture process zone in front of the crack tip is modeled with a potential-based intrinsic cohesive zone approach. Here, SFNM is used to represent the kinematics of crack and the crack front inside the domain without the requirement of remeshing and discontinuous enrichment functions during crack growth, hence resolves the issues associated with the existing discrete numerical methods. A strain smoothing technique is adopted over the domain through which classical domain integration changes to line integration along each boundary of the smoothing cell, hence derivative of shape functions are not required in the computation of the field gradients, thus resolves the issue of element distortion. The proposed numerical framework is firstly verified using the patch test of the two-dimensional specimen under mode I and mode II loading conditions and subsequently extended for solving the two-dimensional standard fracture problems. The effectiveness of the proposed framework is checked by comparing the computational results with the available literature results.

A two-level nesting smoothed extended meshfree method for dynamic fracture analysis

A two-level nesting smoothed extended meshfree method (S-XMM-N) is presented for modeling dynamic fracture behaviors in two-dimensional linear elastic solids. The extended meshfree method based on the partition of unity allows for arbitrary discontinuities, but with low computational efficiency needs improvement. Meanwhile, directly introducing the generalized smoothed strain technique into the extended meshfree method can save more computational times, and with low accuracy also improvement. The proposed method combined the best feature of both the recently developed two-level nesting smoothing meshfree method and the extended radial point interpolation method (XRPIM). On the one hand, the S-XMM-N saves much more computational time than XRPIM by using the proposed two-level nesting triangular sub-domains smoothing integration, where only fewer integration points are employed and the derivatives of shape functions do not need to be calculated for constructing the stiffness matrix. On the other hand, the computational accuracy can be ensured due to the employment of Richardson extrapolation method. The results for the static and dynamic SIFs are compared with the analytical and other numerical solutions by several benchmark fracture problems.

A cell-based smoothed radial point interpolation method applied to kinematic limit analysis of thin plates

The cell-based smoothed radial point interpolation method in conjunction with a second order cone programming is proposed for kinematic limit analysis of rigid-perfectly plastic thin plates in this paper. The transverse displacement field is interpolated by the radial point interpolation method (RPIM) and no rotational degrees of freedom are involved. The gradient smoothing technique is utilized to construct smoothed curvature field in every smoothing domain and there is no need to compute the second derivatives of shape functions. The rotational boundary conditions are satisfied in the process of curvature smoothing and the translational boundary conditions can be directly enforced without any special treatment. The limit analysis problem of thin plates is formulated by minimizing the dissipation power subject to a set of equality constraints and this minimization problem can be expressed easily as a standard second order cone programming. It is testified from the computational results that the proposed procedure can provide reasonable and satisfactory upper bound limit load multipliers for thin plates.

Application of a Calculation Formula for the Self‐Inductance of an Arbitrary‐Shaped Single‐Layer Coil (Derivation of Shape Functions and Inductance Caluculations for Regular Polygonal Coils)

In the design and production of electrical and electronic devices and circuits, it is desirable to have a calculation formula for obtaining exact inductance values, which are generally fundamental for coils. In this paper, a formula that can be applied to the exact calculations of the self-inductance of arbitrary-shaped single-layer coils is presented, and its application to the several types of coils that are technologically valuable is reported. Specifically, the formula employs a new coefficient corresponding to the Nagaoka coefficient and the shape function determined by the coil. Moreover, it realizes a versatile and convenient inductance calculation method. In addition, as the formula is derived via an analytical method, it has the advantage that it can be calculated with a short-time numerical integration. As examples of the application of the formula, rhombic, race-track, elliptic, and regular polygonal coils were discussed, and the process and result of deriving the shape function and new coefficient for each coil were presented. Furthermore, regarding the calculation formula for regular polygonal coils, we made a regular pentagonal coil and conducted an experiment, and the validity of the calculation formula was confirmed. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

An n-sided polygonal selective smoothed finite element method for nearly incompressible visco-hyperelastic soft materials

In this article, an n-sided polygonal smoothed finite element (nSFEM) is formulated for dynamic analyses of nonlinear problems of visco-hyperelastic materials undergoing large deformations. Two types of smoothing domains are constructed for mesh of n-sided polygonal elements. Using the gradient smoothing technique, the calculation of the strain-displacement matrix requires only the shape function values rather than the derivatives of shape function. The simple averaging point interpolation technique is used to calculate the values of the shape functions on the boundary of different smoothing domains. The Total Lagrangian formulation is used to deal with the nonlinear dynamic analysis of soft materials with finite strain. The polygonal FEM with Wachspress coordinate is also implemented for comparison. To overcome volumetric locking, selective smoothing domain scheme is used to calculate the smoothing nodal internal force. To analyze the time-dependent mechanical behavior of soft materials, a constitutive update algorithm for explicit time-integration is implemented based on the generalized Maxwell visco-hyperelastic model. A number of numerical examples are presented to demonstrate the high precision and super convergence, and robustness of the present nS-FEM for incompressible visco-hyperelastic problems.

A cell-based smoothed finite element method for finite elasticity

In this study, we present a displacement based polygonal finite element method for compressible and nearly-incompressible elastic solids undergoing large deformations in two dimensions. This is achieved by projecting the dilatation strain onto the linear approximation space, within the framework of volume averaged nodal projection method. To reduce the numerical integration burden over polytopes, a linear strain smoothing technique is employed to compute the terms in the bilinear/linear form. The salient features of the proposed framework are: (a) does not require derivatives of shape functions and complex numerical integration scheme to compute the bilinear and linear form and (b) volumetric locking is alleviated by adopting the volume averaged nodal projection technique. The efficacy, convergence properties and accuracy of the proposed framework is demonstrated through four standard benchmark problems.

Smoothed floating node method for modelling 2D arbitrary crack propagation problems

In this work, Floating Node Method (FNM), first developed for fracture modelling of laminate composites, is coupled with cell-wise strain Smoothed Finite Element Method (SFEM) for modelling 2D linear elastic fracture mechanics problems. The proposed method is termed as Smoothed Floating Node Method (SFNM). In this framework, FNM is used to represent the kinematics of crack and the crack front inside the domain without the requirement of remeshing and discontinuous enrichment functions during crack growth. For smoothing, a constant smoothing function is considered over the smoothing domains through which classical domain integration changes to line integration along each boundary of the smoothing cell, hence derivative of shape functions are not required in the computation of the field gradients. The values of stress intensity factor are obtained from the SFNM solution using domain based interaction integral approach. Few standard fracture mechanics problems are considered to check the accuracy and effectiveness of the proposed method. The predictions obtained with the proposed framework improves the convergence and accuracy of the results in terms of the stress intensity factors and energy norms.

Fourth‐order phase field model with spectral decomposition for simulating fracture in hyperelastic material

AbstractWe present a fourth‐order phase field model for fracture behavior simulations of hyperelastic material undergoing finite deformation. Governing equations of the fourth‐order phase field model consist of the biharmonic operator of the phase field, which requires the second‐order derivatives of shape function. Therefore, a 5 × 5 Jacobian matrix of isoparametric transformation is constructed. Neo‐Hooken model and Hencky model are adopted as the material constitutive models. The spectral decomposition of stored strain energy is used to distinguish the contributions of tension and compression, and the corresponding stress tensor and constitutive tensors are derived, and subsequently, the numerical framework of modeling fracture with the fourth‐order phase field model is implemented in details. Several typical numerical examples are conducted to demonstrate the robustness and effectiveness of the fourth‐order phase field model in simulating the fracture phenomenon of rubber‐like materials.

The Point Interpolation Method Using Quadratically Consistent Three-Point Integration Scheme for Free Vibration Analysis

The formulation of quadratically consistent three-point integration scheme based on point interpolant method (PIM-QC3) is proposed for free vibration analysis of two-dimensional solids. Based on the condition with regard to nodal shape function and its derivatives for arbitrary-order approximations, PIM-QC3 is developed using point interpolant method and T6-Scheme for node selection. Thus, complex calculation of inverse matrix at every quadrature points is avoided. And the shape functions including corrected derivatives of nodal shape functions at quadrature points, adopt quadratic basis, meet the so-called differentiation of the approximation consistency as well as the discrete divergence consistency. In addition, the Kronecker delta property is possessed. Some problems for the free vibration of two-dimensional solids are employed to examine the accuracy and efficiency of the PIM-QC3. The numerical results indicate that PIM-QC3 is a very accurate, stable and highly efficient method for the structural analysis of free vibration.

Arbitrary order recursive formulation of meshfree gradients with application to superconvergent collocation analysis of Kirchhoff plates

A general arbitrary order recursive gradient formulation is presented for meshfree approximation. According to this method, an nth order recursive meshfree gradient is formulated as an interpolation of the (n − 1)th order gradients by standard first order meshfree gradients, which finally can be expressed as a successive multiplication of standard first order meshfree gradients. This formulation avoids the complex and costly computation of conventional high order derivatives of meshfree shape functions. One crucial ingredient of the proposed methodology is that the resulting recursive meshfree gradients with a pth degree basis function not only meet the conventional pth order consistency conditions for standard gradients, but also satisfy (p + 1)th to (p + n − 1)th extra high order consistency conditions. This important property leads to superconvergent meshfree collocation algorithms and here we focus on the classical fourth order Kirchhoff plate problems. An accuracy analysis of the proposed recursive gradient meshfree collocation formulation for Kirchhoff plates reveals that superconvergence is simultaneously achieved for both even and odd degrees of basis functions. More specifically, two and four additional orders of accuracy are respectively gained by the proposed method for even and odd degree basis functions, compared with the standard meshfree collocation scheme. Furthermore, the extra high order consistency conditions of recursive meshfree gradients enable superconvergent meshfree collocation analysis of Kirchhoff plates using low order basis functions of less than 4th degree, while the standard meshfree collocation approach requires at least a 4th degree basis function to maintain convergence. The accuracy and efficiency of the proposed methodology are holistically demonstrated by numerical results.

Recovering elastic inclusions by shape optimization methods with immersed finite elements

This article presents a finite element method on a fixed mesh for solving a group of inverse geometric problems for recovering the material interface of a linear elasticity system. A partially penalized immersed finite element method is used to discretize both the elasticity interface problems and the objective shape functionals accurately regardless of the shape and location of the interface. Explicit formulas for both the velocity fields and the shape derivatives of IFE shape functions are derived on a fixed mesh and they are employed in the shape sensitivity framework through the discretized adjoint method for accurately and efficiently computing the gradients of objective shape functions with respect to the parameters of the interface curve. The shape optimization for solving an inverse geometric problem is therefore accurately reduced to a constrained optimization that can be implemented efficiently within the IFE framework together with a standard optimization algorithm. We demonstrate features and advantages of the proposed IFE-based shape optimization method by several typical inverse geometric problems for linear elasticity systems.

Global-element-based free element method for solving non-linear and inhomogeneous heat conduction problems

In this paper, a family of global elements (GEs) are constructed for modeling geometries and representing physical variables, based on a set of complete basis functions formulated in terms of normalized global coordinates. The main benefits of using GEs are that the elemental nodes can be distributed and numbered in an arbitrary manner and the global spatial partial derivatives of geometries and physical variables appearing in the governing equations of engineering problems can be directly derived. Based on the constructed GEs and their spatial derivatives of global shape functions, a simple and robust new numerical method, called as the Global-Element-based FRee Element Method (GEFREM), is proposed for solving general two-dimensional heat conduction problems. GEFREM inherits the advantages of the finite element method, mesh free method and free element method. A detailed description of GEFREM for solving general non-linear and inhomogeneous heat conduction problems will be presented in the paper and a number examples are given to verify the correctness and demonstrate the potential of the proposed method.

The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis

The use of modified couple stress theory to simulate the size-dependent phenomenon of composite laminate microplate is commonly limited to simple boundary conditions and mechanical bending load. The small-scale effects on bending and buckling on composite laminate microplate under complex boundary conditions in thermal environment have not been understood fully in the literature. Hence, this research develops, for the first time, a model to overcome the above limitation through the combination of a new modified couple stress theory and isogeometric analysis (IGA). By solving the governing equation using IGA, the thermal displacement, stress and thermal buckling load for various material length scale parameters are obtained. To satisfy the continuous shear stress condition at the layer interfaces, the equilibrium equations as integrated in-plane stress derivatives over the thickness are imposed. In addition, the non-uniform rational B-splines (NURBS) satisfy the higher-order derivative of shape function using the equilibrium equation. Furthermore, to show the effectiveness of presented model for capturing the size effect on thermal bending and thermal buckling of multi-ply laminate microplate, the influences of fiber orientation, thickness ratio, boundary condition and the variation in material length scale parameter are investigated.

A unified polygonal locking-free thin/thick smoothed plate element

A novel cell-based smoothed finite element method is proposed for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields. The domain is discretized with arbitrary polygons and on each side of the polygonal element, discrete shear constraints are considered to relate the kinematical and the independent shear strains. The plate is made of functionally graded material with effective properties computed using the rule of mixtures. The influence of various parameters, viz., the plate aspect ratio and the material gradient index on the static bending response and the first fundamental frequency is numerically studied. It is seen that the proposed element: (a) has proper rank; (b) does not require derivatives of shape functions and hence no isoparametric mapping required; (c) independent of shape and size of elements and (d) is free from shear locking.

Improved integration scheme for the second-order consistent element-free Galerkin method

Consistent element-free Galerkin (CEFG) methods have improved the accuracy, efficiency and convergence of high-order EFG methods remarkably by developing integration schemes using less sampling points and corrected derivatives of nodal shape functions. However, the computation of these derivatives still takes considerable CPU time because it involves the evaluation of shape functions at relatively more points and the solution of equations. To reduce the evaluation of these functions substantially, an improved integration scheme is proposed for the second-order CEFG method. Moreover, these corrected derivatives are explicitly formulated in terms of the shape functions. In consequence, the computational efficiency of second-order CEFG method is further improved by this scheme. Furthermore, the high accuracy and convergence of the CEFG method is still maintained. Numerical results of elastic examples show that this improved CEFG method is evidently faster than the original and its efficiency approaches that of nesting sub-domains gradient smoothing (NSGS), which was developed recently for second-order meshfree Galerkin methods. An extension to small-strain elastoplasticity is also presented where the proposed method is demonstrated to perform much better than NSGS in elastoplastic computations.

Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains

Mathematical analysis and numerical solutions of problems with unknown shapes or geometrical domains is a challenging and rich research field in the modern theory of the calculus of variations, partial differential equations, differential geometry as well as in numerical analysis. In this series of three review papers, we describe some aspects of numerical solution for problems with unknown shapes, which use tools of asymptotic analysis with respect to small defects or imperfections to obtain sensitivity of shape functionals. In classical numerical shape optimization, the boundary variation technique is used with a view to applying the gradient or Newton-type algorithms. Shape sensitivity analysis is performed by using the velocity method. In general, the continuous shape gradient and the symmetric part of the shape Hessian are discretized. Such an approach leads to local solutions, which satisfy the necessary optimality conditions in a class of domains defined in fact by the initial guess. A more general framework of shape sensitivity analysis is required when solving topology optimization problems. A possible approach is asymptotic analysis in singularly perturbed geometrical domains. In such a framework, approximations of solutions to boundary value problems (BVPs) in domains with small defects or imperfections are constructed, for instance by the method of matched asymptotic expansions. The approximate solutions are employed to evaluate shape functionals, and as a result topological derivatives of functionals are obtained. In particular, the topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, defects, source terms and cracks. This new concept of variation has applications in many related fields, such as shape and topology optimization, inverse problems, image processing, multiscale material design and mechanical modeling involving damage and fracture evolution phenomena. In the first part of this review, the topological derivative concept is presented in detail within the framework of the domain decomposition technique. Such an approach is constructive, for example, for coupled models in multiphysics as well as for contact problems in elasticity. In the second and third parts, we describe the first- and second-order numerical methods of shape and topology optimization for elliptic BVPs, together with a portfolio of applications and numerical examples in all the above-mentioned areas.