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– Numerical solution of Plateau’s problem of minimal surface using non-variational finite element method. The paper aims to discuss this issue.
Abstract
Purpose
Numerical solution of Plateau’s problem of minimal surface using non-variational finite element method. The paper aims to discuss this issue.
Design/methodology/approach
An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.
Findings
The solutions obtained here are examined for different cases of non-linearity and are found sufficiently accurate.
Originality/value
The manuscript provide the non-variational solution for Plateau’s problem. Thus it has a good value in engineering application.
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Keywords
Kaifeng Jiang, Si Yuan and Qinyan Xing
This paper aims to propose a new adaptive strategy for two-dimensional (2D) nonlinear finite element (FE) analysis of the minimal surface problem (MSP) based on the element energy…
Abstract
Purpose
This paper aims to propose a new adaptive strategy for two-dimensional (2D) nonlinear finite element (FE) analysis of the minimal surface problem (MSP) based on the element energy projection (EEP) technique.
Design/methodology/approach
By linearizing nonlinear problems into a series of linear problems via the Newton method, the EEP technique, which is an effective and reliable point-wise super-convergent displacement recovery strategy for linear FE analysis, can be directly incorporated into the solution procedure. Accordingly, a posteriori error estimate in maximum norm was established and an adaptive 2D nonlinear FE strategy of h-version mesh refinement was developed.
Findings
Three classical known surfaces, including a singularity problem, were analysed. Moreover, an example whose analytic solution is unavailable was considered and a comparison was made between present results and those computed by the MATLAB PDE toolbox. The results show that the adaptively-generated meshes reflect the difficulties inherent in the problems and the proposed adaptive analysis can produce FE solutions satisfying the user-preset error tolerance in maximum norm with a fair adaptive convergence rate.
Originality/value
The EEP technique for linear FE analysis was extended to the nonlinear procedure of MSP and can be expected to apply to other 2D nonlinear problems. The employment of the maximum norm makes point-wisely error control on the sought surfaces possible and makes the proposed method distinguished from other adaptive FE analyses.
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Mehdi Dehghan and Vahid Mohammadi
This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a…
Abstract
Purpose
This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.
Design/methodology/approach
First, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.
Findings
This study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.
Originality/value
This study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.
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Haohan Sun and Si Yuan
A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique.
Abstract
Purpose
A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique.
Design/methodology/approach
By linearizing the free vibration problem of elastic membranes into a series of linear equivalent problems, reliable a posteriori point-wise error estimator is constructed via EEP super-convergent technique. Hierarchical local mesh refinement is incorporated to better deal with tough problems.
Findings
Several classical examples were analyzed, confirming the effectiveness of the EEP-based error estimation and overall adaptive procedure equipped with a local mesh refinement scheme. The computational results show that the adaptively-generated meshes reasonably catch the difficulties inherent in the problems and the procedure yields both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm.
Originality/value
By reasonable linearization, the linear-problem-based EEP technique is successfully transferred to two-dimensional eigenproblems with local mesh refinement incorporated to effectively and flexibly deal with singularity problems. The corresponding adaptive strategy can produce both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm and thus can be expected to apply to other types of eigenproblems.
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Keywords
Haohan Sun and Si Yuan
An improved adaptive finite element analysis based on local error estimate is proposed via the element energy projection (EEP) technique. This paper aims to discuss the…
Abstract
Purpose
An improved adaptive finite element analysis based on local error estimate is proposed via the element energy projection (EEP) technique. This paper aims to discuss the aforementioned idea.
Design/methodology/approach
The computational region for a posteriori error estimation based on EEP method is further confined to a critical set of local elements generated in the previous adaptive step, enhancing efficiency while maintaining accuracy. The adaptive procedure incorporated with hierarchical mesh refinement is then developed.
Findings
The effectiveness of the improved error estimation of the overall adaptive analysis is confirmed by several benchmark examples. The results show that the shrinkage of the local computational region has little negative influence on the accuracy of a posteriori error estimation, thus yielding an improved adaptive procedure with simplified logic and reduced cost.
Originality/value
By localizing the computational region for error estimation, two crucial but cumbersome tricks, i.e. treatments of virtual elements and hanging nodes, are removed, giving the proposed approach full clarity and flexibility. The improved adaptive procedure characterizes simpler and faster computational algorithm and can produce results with required accuracy measured in maximum norm.
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Zhuo‐Jia Fu, Qing‐Hua Qin and Wen Chen
The purpose of this paper is to develop a hybrid‐Trefftz (HT) finite element model (FEM) for simulating heat conduction in nonlinear functionally graded materials (FGMs) which can…
Abstract
Purpose
The purpose of this paper is to develop a hybrid‐Trefftz (HT) finite element model (FEM) for simulating heat conduction in nonlinear functionally graded materials (FGMs) which can effectively handle continuously varying properties within an element.
Design/methodology/approach
In the proposed model, a T‐complete set of homogeneous solutions is first derived and used to represent the intra‐element temperature fields. As a result, the graded properties of the FGMs are naturally reflected by using the newly developed Trefftz functions (T‐complete functions in some literature) to model the intra‐element fields. The derivation of the Trefftz functions is carried out by means of the well‐known Kirchhoff transformation in conjunction with various variable transformations.
Findings
The study shows that, in contrast to the conventional FEM, the HT‐FEM is an accurate numerical scheme for FGMs in terms of the number of unknowns and is insensitive to mesh distortion. The method also performs very well in terms of numerical accuracy and can converge to the analytical solution when the number of elements is increased.
Originality/value
The value of this paper is twofold: a T‐complete set of homogeneous solutions for nonlinear FMGs has been derived and used to represent the intra‐element temperature; and the corresponding variational functional and the associated algorithm has been constructed.
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Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community…
Abstract
Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.
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To focus on grid generation which is an essential part of any analytical tool for effective discretization.
Abstract
Purpose
To focus on grid generation which is an essential part of any analytical tool for effective discretization.
Design/methodology/approach
This paper explores the application of the possibility of unstructured triangular grid generation that deals with derivationally continuous, smooth, and fair triangular elements using piecewise polynomial parametric surfaces which interpolate prescribed R3 scattered data using spaces of parametric splines defined on R2 triangulations in the case of surfaces in engineering sciences. The method is based upon minimizing a physics‐based certain natural energy expression over the parametric surface. The geometry is defined as a set of stitched triangles prior to the grid generation. As for derivational continuities between the two triangular patches C0 and C1 continuity or both, as per the requirements, has been imposed. With the addition of a penalty term, C2 (approximate) continuity can also be achieved. Since, in this work physics‐based approach has been used, the grid is analyzed using intersection curves with three‐dimensional planes, and intrinsic geometric properties (i.e. directional derivatives), for derivational continuity and smoothness.
Findings
The triangular grid generation that deals with derivationally continuous, smooth, and fair triangular elements has been implemented in this paper for surfaces in engineering sciences.
Practical implications
This paper deals with the important problem of grid generation which is an essential part of any analytical tool for effective discretization. And, the examples to demonstrate the theoretical model of this paper have been chosen from different branches of engineering sciences. Hence, the results of this paper are of practical importance for grid generation in engineering sciences.
Originality/value
The paper is theoretical with worked examples chosen from engineering sciences.
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Mingkang Zhang, Yongqiang Yang, Wentao Qin, Shibiao Wu, Jie Chen and Changhui Song
This study aims to focus on the optimized design and mechanical properties of gradient triply periodic minimal surface cellular structures manufactured by selective laser melting.
Abstract
Purpose
This study aims to focus on the optimized design and mechanical properties of gradient triply periodic minimal surface cellular structures manufactured by selective laser melting.
Design/methodology/approach
Uniform and gradient IWP and primitive cellular structures have been designed by the optimized function in MATLAB, and selective laser melting technology was applied to manufacture these cellular structures. Finite element analysis was applied to optimize the pinch-off problem, and compressive tests were carried out for the evaluation of mechanical properties of gradient cellular structures.
Findings
Finite element analysis shows that the elastic modulus of IWP increased as design parameter b increased, and then decreased when parameter b is higher than 5.5. The highest elastic modulus of primitive increased by 89.2% when parameter b is 6. The compressive behavior of gradient IWP and primitive shows a layer-by-layer way, and elastic modulus and first maximum compressive strength of gradient primitive are higher than that of gradient IWP. The effective energy absorption of gradient cellular structures increased as the average porosity decreased, and the effective energy absorption of gradient primitive is about twice than that of gradient IWP.
Originality/value
This paper presents an optimized design method for the pinch-off problem of gradient triply periodic minimal surface cellular structures.
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Examines the tenth published year of the ITCRR. Runs the whole gamut of textile innovation, research and testing, some of which investigates hitherto untouched aspects. Subjects…
Abstract
Examines the tenth published year of the ITCRR. Runs the whole gamut of textile innovation, research and testing, some of which investigates hitherto untouched aspects. Subjects discussed include cotton fabric processing, asbestos substitutes, textile adjuncts to cardiovascular surgery, wet textile processes, hand evaluation, nanotechnology, thermoplastic composites, robotic ironing, protective clothing (agricultural and industrial), ecological aspects of fibre properties – to name but a few! There would appear to be no limit to the future potential for textile applications.
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