Adjustable Piecewise Quartic Hermite Spline Curve with Parameters

In this paper, the quartic Hermite parametric interpolating spline curves are formed with the quartic Hermite basis functions with parameters, the parameter selections of the spline curves are investigated, and the criteria for the curve with the shortest arc length and the smoothest curve are given. When the interpolation conditions are set, the proposed spline curves not only achieve C1-continuity but also can realize shape control by choosing suitable parameters, which addressed the weakness of the classical cubic Hermite interpolating spline curves.

Couple stress-based unsymmetric 8-node planar membrane elements with good tolerances to mesh distortion

PurposeThe paper aims to propose two new 8-node quadrilateral membrane elements with good distortion tolerance for the modified couple stress elasticity based on the unsymmetric finite element method (FEM).Design/methodology/approachThe nodal rotation degrees of freedom (DOFs) are introduced into the virtual work principle and constrained by the penalty function for approximating the test functions of the physical rotation and curvature. Therefore, only the C0 continuity instead of C1 continuity is required for the displacement during the element construction. The first unsymmetric element assumes the test functions of the displacement and strain using the standard 8-node isoparametric interpolations, while these test functions in the second model are further enhanced by the nodal rotation DOFs. Besides, the trial functions in these two elements are constructed based on the stress functions that can a priori satisfy related governing equations.FindingsThe benchmark tests show that both the two elements can efficiently simulate the size-dependent plane problems, exhibiting good numerical accuracies and high distortion tolerances. In particular, they can still exactly reproduce the constant couple stress state when the element shape deteriorates severely into the degenerated triangle. Moreover, it can also be observed that the second element model, in which the linked interpolation technique is used, has better performance than the first one, especially in capturing the steep gradients of the physical rotations.Originality/valueAs the proposed new elements use only three DOFs per node, they can be readily incorporated into the existing finite element (FE) programs. Thus, they are of great benefit to analysis of size-dependent membrane behaviors of micro/nano structures.

Investigation of layered composite plates under acoustic emission using an appropriate FE model

Structural damages generate acoustic emission (AE) in the media and cause extensional and flexural acoustic waves. Often in structures like plates, flexural mode is predominant. In this study, the flexural mode AE waveforms due to simulated damage are studied for multi-layered composite plates. A generalized refined 2D plate theory, which satisfies the transverse shear stress continuity at the layer interfaces, is proposed here for modelling the plates. This formulation is implemented through finite element method where a four-node rectangular element, that satisfies C1 continuity, is used. Plates, having different thickness ratios, are studied through numerical examples using the model. Results are validated wherever applicable and some new results are obtained. The results indicate that the proposed model can simulate the flexural waveforms realistically for ‘very thin’ to ‘moderately thick’ plates. It is also found that the present model is as accurate as 3DFEM but it possesses much better computational efficiency.

A Transitional Connection Method for the Design of Functionally Graded Cellular Materials

In recent years, the functionally graded materials (FGM) with cellular structure have become a hot spot in the field of materials research. For the continuously varying cellular structure in the layer-wise FGM, the connection of gradient cellular structures has become the main problem. Unfortunately, the effect of gradient connection method on the overall structural performance lacks attention, and the boundary mismatch has enormous implications. Using the homogenization theory and the level set method, this article presents an efficient topology optimization method to solve the connection issue. Firstly, a simple but efficient hybrid level set scheme is developed to generate a new level set surface that has the partial features of two candidate level sets. Then, when the new level set surface is formed by considering the level set functions of two gradient base cells, a special transitional cell can be constructed by finding the zero level set of this generated level set surface. Since the transitional cell has the geometric features of two gradient base cells, the shape of the transitional cell fits perfectly with its connected gradient cells on both sides. Thus, the design of FGM can have a smooth connectivity with C1 continuity without any complex numerical treatments during the optimization. A number of examples on both 2D and 3D are provided to demonstrate the characteristics of the proposed method. Finite element simulation has also been employed to calculate the mechanical properties of the designs. The simulation results show that the FGM devised by the proposed method exhibits better mechanical performances than conventional FGM with only C0 continuity.

Static analysis of FG plates using T-splines based isogeometric approach and a refined plate theory

In this study, a novel refined plate theory (RPT) is developed for the geometrically linear static analysis of FG plates, which is a simplification of the higher-order shear deformation theories (HSDTs). It improves the computational efficiency while preserving the accuracy advantage of HSDTs. The C1-continuity problem is overcome by isogeometric analysis (IGA), which shows more advantages than the C0 elements based finite element analysis. By T-splines, the computational cost is effectively reduced, since compared to NURBS based IGA, T-splines can achieve local refinement and improve the utilization of control points. The rule of mixture with power-law and Mori–Tanaka scheme are adopted to calculate the material properties of the plate. Several numerical experiments are given to prove the efficiency of the proposed method

Quadratic Approximation of Cubic Curves

We present a simple degree reduction technique for piecewise cubic polynomial splines, converting them into piecewise quadratic splines that maintain the parameterization and C1 continuity. Our method forms identical tangent directions at the interpolated data points of the piecewise cubic spline by replacing each cubic piece with a pair of quadratic pieces. The resulting representation can lead to substantial performance improvements for rendering geometrically complex spline models like hair and fiber-level cloth. Such models are typically represented using cubic splines that are C1-continuous, a property that is preserved with our degree reduction. Therefore, our method can also be considered a new quadratic curve construction approach for high-performance rendering. We prove that it is possible to construct a pair of quadratic curves with C1 continuity that passes through any desired point on the input cubic curve. Moreover, we prove that when the pair of quadratic pieces corresponding to a cubic piece have equal parametric lengths, they join exactly at the parametric center of the cubic piece, and the deviation in positions due to degree reduction is minimized.

EFFECT OF POROSITY ON BEHAVIOURS OF PLATE STRUCTURES

In this paper, effect of porosity on nonlinear analysis of plate structures is presented. Two porous distributions are considered. Governing equations are expressed by using isogeometric analysis (IGA) and the third-order shear deformation theory (TSDT). With these approaches, it is easy to fulfil the C1-continuity requirement of the plate model. The obtained results demonstrate the significance of porosity volume fraction, porosity distributions and volume fraction exponent on nonlinear analysis of the plate structures.

Construction of Cubic Timmer Triangular Patches and its Application in Scattered Data Interpolation

This paper discusses scattered data interpolation by using cubic Timmer triangular patches. In order to achieve C1 continuity everywhere, we impose a rational corrected scheme that results from convex combination between three local schemes. The final interpolant has the form quintic numerator and quadratic denominator. We test the scheme by considering the established dataset as well as visualizing the rainfall data and digital elevation in Malaysia. We compare the performance between the proposed scheme and some well-known schemes. Numerical and graphical results are presented by using Mathematica and MATLAB. From all numerical results, the proposed scheme is better in terms of smaller root mean square error (RMSE) and higher coefficient of determination (R2). The higher R2 value indicates that the proposed scheme can reconstruct the surface with excellent fit that is in line with the standard set by Renka and Brown’s validation.