Serendipity Elements Research Articles

Marketing Mother Nature’s Molecules

Natural products research long ago fell out of vogue at big pharma firms, which have turned to speedier, more automated ways to discover new drugs. Yet molecules made by Mother Nature, or derivatives thereof, still account for nearly half of the drugs on the market. The recently established Natural Products Discovery Institute (NPDI), based in Doylestown, Pa., hopes to keep natural products in the new-drug pipeline by opening access to the huge repository of microbe and plant samples amassed over the decades at Merck & Co. and its 2009 acquisition, Schering-Plough. The creation of NPDI, like many fruitful scientific endeavors, had an element of serendipity. After abandoning natural products research in 2008, Merck wanted to find a new home for its extensive library. The Merck employee tasked with the job reached out to many for-profit entities around the globe but found little interest. He then decided to try the Hepatitis ...

A 3D pyramid spline element

In this paper, a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method, which achieves the second order completeness in Cartesian coordinates. Some appropriate examples were employed to evaluate the performance of the proposed element. The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted, and it is comparable to the Lagrange element Q27. It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.

Chance and Serendipity in Science: Two Examples from My Own Career

The usual scientific paper follows a rather narrowly (but not ever rigidly) defined pattern. Both the author and the journal like to see a linear logical presentation of a “story.” Seldom does the paper give the reader the “backstory.” Where did the idea come from in the first place? How many false leads led down blind alleys? What happened by chance and what by logical planning? Was there an element of serendipity involved? Perhaps as we enter the paperless era and do not have to count words quite so religiously, it may be possible to encourage a more freewheeling scientific paper, but for now, we have to rely on the historians of science and/or those who “tell all” about their own research. “Reflections” seems an appropriate space for the latter. I have chosen two scenarios from my own career in which happy accidents played important roles but, unhappily, received little recognition in my published papers.

Correction of node mapping distortions using universal serendipity elements in dynamical problems

In this paper, the use of universal serendipity elements (USE) to eliminate node mapping distortions for dynamic problem is presented. Rectangular shaped elements for USE are being introduced by using a flexible master element with an adjustable edge node location. The shape functions of the universal serendipity formulation are used to derive the mass and damping matrices for the dynamic analyses. These matrices eliminate the node mapping distortion errors that occurs incase of the standard shape function formulations. The verification of new formulation will be tested and the errors encountered in the standard formulation will be studied for a dynamically loaded deep cantilever.

A 3D hexahedral spline element

Based on the tetrahedral volume coordinates and the B-net method, a 3D hexahedral spline element with 21 nodes is proposed, which includes an internal node and achieves the second order completeness in the Cartesian coordinates. Some numerical examples are employed to evaluate the performance of the proposed element. The numerical results show that the 21-node element has much better performance than that of the isoparametric 20-node Serendipity element especially for the distorted meshes, and it is comparable to the 27-node Lagrange element. It has also been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.

Convergence Analysis for Cubic Serendipity Finite Elements with Thirty-Two Degrees of Freedom

In this paper, we consider the Poisson problem with homogeneous Dirichlet boundary conditions on the bounded open domain and analyze convergence phenomena that appear by using cubic serendipity block elements with thirty-two degrees of freedom. By means of the standard Legendre polynomial technique, we derive an error estimate, which is not superconvergent and shows this serendipity element has not pointwise supercloseness properties.

Influence of cracked concrete models on the nonlinear analysis of High Strength Steel Fibre Reinforced Concrete corbels

In the present study, new material constitutive relationships for High Strength Steel Fibre Reinforced Concrete (HSSFRC) have been formulated by means of a regression analysis of many experimental data (available in literature) by using SPSS (Statistical Package for the Social Sciences) program. These proposed material constitutive relationships have been employed for formulating the material finite element models to study the behaviour of HSSFRC corbels. The corbel is considered in a plane stress state and an eight-node serendipity element has been employed to represent the concrete, a strain hardening plasticity approach is used to model the compression behaviour, the yield condition is formulated in terms of the first two stress invariants, a tensile strength criterion is used to predict crack initiation, two tensile stress degradation functions are used to model the tension stiffening. Five different cracked shear moduli have been compared in this paper. For plain high strength reinforced concrete corbels, the first tension stiffening ( TS 1) model with cracked shear modulus model ( G ¯ 4 ) and the second tension stiffening ( TS 2) with model ( G ¯ 2 ) give good results compared with experimental results. For HSSFRC corbels, the second tension stiffening ( TS 2) model with cracked shear modulus models ( G ¯ 2 ) and ( G ¯ 3 ) give good results compared with experimental results. The numerical results of the proposed material models and corbels behaviour compared with published experimental results showed good agreement.

Vibration Analysis of Thick Arbitrarily Laminated Plates of Various Shapes and Edge Conditions

Free vibration analysis of arbitrarily laminated plates of quad, penta and hexagonal shapes, which have combinations of clamped, simply supported and free edge conditions is performed. The finite element formulation is based on first and higher order shear deformation theories to study the free vibration response of thick laminated composite plates. A finite element code is developed incorporating shear deformation theories using an 8-noded serendipity element. The effect of plate shape, arbitrary lamination and different edge conditions on natural frequencies and mode shapes are investigated. A systematic study is carried out to determine the influence of material orthotropy and aspect ratio on free vibration response. For various cases, the comparisons of results from present study showed good agreement with those published in the literature.

Modelling and numerical simulation of high strength fibre reinforced concrete corbels

In the present study, new constitutive models for high strength steel fibre reinforced concrete (HSSFRC) have been formulated by means of a regression analysis of many experimental data (from literature) by using SPSS-statistical program. This proposed constitutive models have been employed for formulating the material finite element models to study the behaviour of HSSFRC corbels. The corbel is considered in a plane stress state and eight-node Serendipity element has been employed to represent the concrete, a strain hardening plasticity approach is used to model the compression behaviour, the yield condition is formulated in terms of the first two stress invariants, a tensile strength criterion is used to predict crack initiation. The numerical results of proposed material models and corbels behaviour compared with published experimental results showed good agreement.

Reliable second‐order hexahedral elements for explicit methods in nonlinear solid dynamics

AbstractSecond‐order hexahedral elements are common in static and implicit dynamic finite element codes for nonlinear solid mechanics. Although probably not as popular as first‐order elements, they can perform better in many circumstances, particularly for modeling curved shapes and bending without artificial hourglass control or incompatible modes. Nevertheless, second‐order brick elements are not contained in typical explicit solid dynamic programs and unsuccessful attempts to develop reliable ones have been reported. In this paper, 27‐node formulations, one for compressible and one for nearly incompressible materials, are presented and evaluated using non‐uniform row summation mass lumping in a wide range of nonlinear example problems. The performance is assessed in standard benchmark problems and in large practical applications using various hyperelastic and inelastic material models and involving very large strains/deformations, severe distortions, and contact‐impact. Comparisons are also made with several first‐order elements and other second‐order hexahedral formulations. The offered elements are the only second‐order ones that performed satisfactorily in all examples, and performed generally at least as well as mass lumped first‐order bricks. It is shown that the row summation lumping is vital for robust performance and selection of Lagrange over serendipity elements and high‐order quadrature rules are more crucial with explicit than with static/implicit methods. Whereas the reliable performance is frequently attained at significant computational expense compared with some first‐order brick types, these elements are shown to be computationally competitive in flexure and with other first‐order elements. These second‐order elements are shown to be viable for large practical applications, especially using today's parallel computers. Published in 2010 by John Wiley & Sons, Ltd.

Eigenanalysis of Two-Dimensional Acoustic Cavities Using Transfinite Interpolation

This paper discusses the efficient extraction of eigenfrequencies in two-dimensional acoustic cavities using higher order finite elements, called “Coons-patch macroelements”. The acoustic pressure is approximated within the entire domain using the bivariate transfinite (Coons-Gordon) interpolation formula. Basically, the proposed macroelements constitute a generalization of the well-known Serendipity and Lagrangian type elements. The paper investigates the performance of the proposed methodology in several examples and definitely correlates the necessity of using internal nodes or domain decomposition with the percentage of open boundaries over the entire boundary as well as their relative position. It was found that for simple shapes, the entire acoustic cavity can be considered as a single quadrilateral patch, i.e. as one Coons-patch macroelement, while for complex shapes the cavity should be divided into a small number of subregions.

Geometrically Nonlinear Finite Element Analysis of Composite Plates Integrated with the Patches of Piezoelectric Fiber-reinforced Composite

Geometrically nonlinear static analysis of laminated smart composite plates integrated with the patches of piezoelectric fiber-reinforced composite (PFRC) material acting as the distributed actuators has been carried out by a generalized energy based finite element model. The Von Karman type nonlinear strain displacement relations and a simple first-order shear deformation theory are used for deriving this coupled electromechanical finite element model. Eight-noded isoparametric serendipity elements are used for discretizing the overall plates. The nonlinear finite element equilibrium equations are solved by Picard's direct iteration method under relaxation. The performance of the PFRC patches has been investigated for both symmetric and antisymmetric cross-ply and antisymmetric angle-ply laminated composite plate substrates with different boundary conditions. Particular emphasis has been placed on investigating the effect of variation of piezoelectric fiber orientation on the actuating capability of the PFRC patches for counteracting the nonlinear deformations of the smart composite plates. The results suggest the potential use of the patches of PFRC material for distributed control of nonlinear deformations of smart composite structures.

Effects of element distortions on the performance of enriched quadrilateral elements

It is well known that the accuracy of finite element solutions deteriorate in the presence of severe mesh distortions. But distortion is often unavoidable in mesh procedure involving complex geometry. Lee and Bathe (1993) studied the influence of mesh distortion on the serendipity and Lagrange quadrilateral elements. Lautersztajn and Samuelsson (2000) discussed the effects of geometric distortions on four-node isoparametric quadrilateral elements and concluded that the element performance can be rendered ?insensitive? to a particular type of mesh distortion by increasing the order of the interpolation functions for the displacement field. In order to overcome the influence of element distortions, unsymmetric 8-node (Rajendran and Liew 2003) and unsymmetric 20-node element (Ooi et al. 2004) are developed to reproducing any linear and quadratic displacement field under any admissible mesh distortions. However, they will produce an asymmetrical stiffness matrix, so these formulations require an asymmetrical solver to solve the resulting stiffness equations. The goal of this paper is to discuss the effects of element distortions on the accuracy and efficiency of enriched quadrilateral elements with bubble functions. A bubble function is defined as a function that vanishes along the element boundaries. Bubble functions have been introduced to construct plate element models (Auricchio and Taylor 1995, Cook et al. 2002, Hong et al. 2001). They are employed to solve advection-diffusion problems by Brezzi, Franca and Farhat (Brezzi et al. 1992, Brezzi and Russo 1994, Franca and Farhat 1995). Furthermore, the limitation of bubble functions is discussed by Franca and Farhat (1994) and error analysis of residual-free bubbles is discussed by Brezzi et al. (1999) and Sangalli (2000).

THE USE OF ENRICHED HEXAHEDRAL ELEMENTS WITH BUBBLE FUNCTIONS FOR FINITE ELEMENT ANALYSIS

In this paper, the concept that adds the interior nodes of the Lagrange elements to the serendipity elements is described and a family of enriched elements is presented to improve the accuracy of finite element analysis. By the use of the static condensation technique at the element level, the extra computation time in using these elements can be ignored. Three-dimensional elastic problems are used as examples in this paper. The numerical results show that these enriched elements are more accurate than the traditional serendipity elements. The convergence rate of the enriched elements is the same as the traditional serendipity elements. In the numerical example, the error norm of the first order enriched elements can be reduced when compared with the use of the traditional serendipity element, but the computation time is increased a little. The use of enriched second and third order hexahedral elements does not only improve accuracy, but also saves the computation time for solving the system of equations, when the precondition conjugate gradient method is used to solve the system of equations. The saving of computation time is due to the decrease in the number of iteration for the iteration method.

The use of 2D enriched elements with bubble functions for finite element analysis

In this paper, the concept that adds the interior nodes of the Lagrange elements to the serendipity elements is described and a family of enriched elements is presented to improve the accuracy of finite element analysis. By the use of the static condensation technique at the element level, the extra computation time in using these elements can be ignored. Plane stress problems are used as examples in this paper. The numerical results show that these enriched elements are more accurate than the traditional serendipity elements. The convergence rate of the proposed elements is the same as the traditional serendipity elements. The error norm of the second and third order proposed elements can be reduced from 40% to 60% when compared with the use of the traditional serendipity elements. In the numerical examples, the use of the second and third order proposed elements not only give an improvement in element accuracy but also save computation time, when the precondition conjugate gradient method is used to solve the system of equations. The saving of computation time is due to the decrease of iteration number in iteration method.

탄성지지된 집중질량을 갖는 변단면 후판의 진동해석

This study is undertaken for the vibration analysis of tapered thick plate with concentrated mass on elastic foundation. The boundary condition of the plate is analyzed with the 4-sides simply supported and 4-fixed basis. This study find out the frequency following the change in size for each foundational variable on Pasternak foundation, one of the two-parameter elastic foundation parameter that considered the shear layer to the Winkler foundation parameter. The concentrated mass is applied with the consideration of mass of the entire plate, and the change of frequency is studies on each location with the consideration of reacting for the three locations for concentrated mass. And, in order to find out the change of frequency on the thickness of the plate, it considered tapered ratio that linearly changes depending on the length of the plate with the thickness of the plate in x-direction, and the tapered ratio has changes with 4 types (<TEX>$\alpha$</TEX>=0.25, 0, 5, 0.75, and 1.0). For the interpretation, the program using finite element method (F.E.M.) is used and the element coordination is used the 8-node serendipity element. Therefore, the purpose of this study is to find out the characteristics of plate vibration under the mechanica vibration or external vibration factor to facilitate as the basic data of the design to secure the stability.

On the finite element method on quadrilateral meshes

The theoretical analysis of the finite element method is well established in the case of triangular or tetrahedral meshes. In this case optimal approximation properties have been proved in all reasonable functional norms. Several commercial codes use this method which is now in the common practice of engineering applications. On the other hand, the case of quadrilateral or hexahedral meshes, even if commonly used in the applications (sometimes it seems to be even more popular than the previous one), has not been studied in such deep detail, probably because it hides some insidious issues. For simplicity, we restrict our analysis to the two-dimensional case; three-dimensional analysis in some cases is a straightforward extension, in some others is more complicated. Indeed, we shall show that some commonly used quadrilateral finite elements present a lack of convergence when general regular meshes are used. The list of suboptimal elements includes serendipity (or trunk) elements, face elements, edge elements. A rigorous theory is presented, which gives necessary and sufficient conditions for optimal order approximation. Our theory is supported by several numerical experiments, which are taken from various engineering applications, ranging from elasticity and fluid dynamics to acoustics and electromagnetics. Example of suboptimal elements include: 8-node element for Poisson problem, Q 2 – P 1 Stokes element, face elements for acoustic problem, edge elements for electromagnetic problems.

Free vibration analysis of two-dimensional structures using Coons-patch macroelements

Recently, Coons’ interpolation was used for the construction of large finite elements with degrees of freedom appearing mostly along the boundaries of a structure. In the regime of elasticity problems, these so-called “Coons-patch macroelements” were successfully applied to the static analysis of plane structures [C.G. Provatidis, Analysis of axisymmetric structures using Coons’ interpolation, Finite Elem. Anal. Des. 39 (2003) 535–558.] while this paper continues the research by investigating their performance in the extraction of natural frequencies and mode shapes. Apart from the piecewise-linear and cubic B-splines interpolation previously used, the performance of Lagrange polynomials and the role of additional internal nodes is studied here. Relationships with classical Serendipity and Lagrangian type elements are discussed. Moreover, the capability of Coons-patch macroelements to couple with conventional finite elements is investigated. The proposed method was applied to three illustrative examples and it was successfully compared with conventional bilinear finite elements.

Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory

A two-dimensional finite element model is presented to perform the linear static analysis of laminated orthotropic composite plates based on a refined higher order shear deformation theory. The theory accounts for parabolic distributions of transverse shear stresses and requires no shear correction factors. A finite element program is developed using serendipity element with seven degrees of freedom per node. The present solutions are compared with those obtained using three-dimensional elasticity theory and those obtained by other researchers. The theory accurately predicts displacements and transverse shear stresses compared to previously developed theories for thick plates and are very close to three-dimensional elasticity solutions. The effects of transverse shear deformation, material anisotropy, aspect ratio, fiber orientation and lamination sequence on transverse shear stresses are investigated. The error in values of transverse shear stresses decreases as the number of lamina increases, for a plate of same thickness. An increase in degree of anisotropy results in lower values of deflection in the plate. For cross-ply plate an increase in anisotropy results in an increase in effective stress whereas for angle-ply plate the effect is almost negligible. Through thickness variation of transverse shear stresses are independent of anisotropy. The maximum effective stress increases exponentially at lower values of anisotropy and reaches to an asymptotic value at higher values. The stacking sequence has a significant effect on the transverse deflections and shear stress. Rectangular plates experience less effective, in-plane and transverse shear stresses compared to square plates.

Adaptive mesh refinement FEM for damage evolution of heterogeneous brittle media

Damage evolution of heterogeneous brittle media involves a wide range of length scales. The coupling between these length scales is the underlying mechanism of damage evolution and rupture. However, few of previous numerical algorithms consider the effects of the trans-scale coupling effectively. In this paper, an adaptive mesh refinement finite element method (FEM) algorithm is developed to simulate this trans-scale coupling. The adaptive serendipity element is implemented in this algorithm, and several special discontinuous base functions are created to avoid the incompatible displacement between the elements. Both the benchmark and a typical numerical example under quasi-static loading are given to justify the effectiveness of this model. The numerical results reproduce a series of characteristics of damage and rupture in heterogeneous brittle media.