An h-version adaptive FEM for eigenproblems in system of second order ODEs: vector Sturm-Liouville problems and free vibration of curved beams

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yongliang Wang
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
High Precision ◽  
Numerical Solutions ◽  
Rayleigh Quotient ◽  
Second Order ◽  
Adaptive Finite Element Method ◽  
Multiple Cracks ◽  
Content Type ◽  
Sturm Liouville

Purpose This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions. Design/methodology/approach This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error. Findings Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance. Originality/value The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

scholarly journals Adaptive finite element analysis for damage detection of non-uniform Euler–Bernoulli beams with multiple cracks based on natural frequencies

2018 ◽  
Vol 35 (3) ◽  
pp. 1203-1229 ◽  
Author(s):  
Yongliang Wang ◽  
Yang Ju ◽  
Zhuo Zhuang ◽  
Chenfeng Li
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Damage Detection ◽  
Crack Detection ◽  
Iteration Process ◽  
Adaptive Finite Element Method ◽  
Multiple Cracks ◽  
Adaptive Finite Element ◽  
Content Type ◽  
Vibration Problems

Purpose This study aims to develop an adaptive finite element method for structural eigenproblems of cracked Euler–Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems). Design/methodology/approach The weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler–Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, using the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton–Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems. Findings Numerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler–Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method. Originality/value The proposed combination of methodologies described in the paper leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

scholarly journals Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique

2017 ◽  
Vol 34 (8) ◽  
pp. 2862-2876 ◽  
Author(s):  
Si Yuan ◽  
Kangsheng Ye ◽  
Yongliang Wang ◽  
David Kennedy ◽  
Frederic W. Williams
Keyword(s):  
Finite Element ◽  
Fourth Order ◽  
Variable Coefficients ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Fe Method ◽  
Eigenvalues And Eigenfunctions ◽  
Content Type ◽  
Sturm Liouville ◽  
Element Energy Projection

Purpose The purpose of this paper is to present a numerically adaptive finite element (FE) method for accurate, efficient and reliable eigensolutions of regular second- and fourth-order Sturm–Liouville (SL) problems with variable coefficients. Design/methodology/approach After the conventional FE solution for an eigenpair (i.e. eigenvalue and eigenfunction) of a particular order has been obtained on a given mesh, a novel strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the element energy projection (EEP) technique for linear problems to calculate the super-convergent FE solutions for eigenfunctions anywhere on any element. These EEP super-convergent solutions are used to estimate the FE solution errors and to guide mesh refinements, until the accuracy matches user-preset error tolerance on both eigenvalues and eigenfunctions. Findings Numerical results for a number of representative and challenging SL problems are presented to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method. Research limitations/implications The method is limited to regular SL problems, but it can also solve some singular SL problems in an indirect way. Originality/value Comprehensive utilization of the EEP technique yields a simple, efficient and reliable adaptive FE procedure that finds sufficiently fine meshes for preset error tolerances on eigenvalues and eigenfunctions to be achieved, even on problems which proved troublesome to competing methods. The method can readily be extended to vector SL problems.

An hp-version adaptive finite element algorithm for eigensolutions of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yongliang Wang ◽  
Jianhui Wang
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Cylindrical Shells ◽  
Higher Order ◽  
Wave Number ◽  
Adaptive Finite Element ◽  
Content Type ◽  
Circular Cylindrical Shells ◽  
Geometrical Factors ◽  
Circumferential Wave

PurposeThis study presents a novel hp-version adaptive finite element method (FEM) to investigate the high-precision eigensolutions of the free vibration of moderately thick circular cylindrical shells, involving the issues of variable geometrical factors, such as the thickness, circumferential wave number, radius and length.Design/methodology/approachAn hp-version adaptive finite element (FE) algorithm is proposed for determining the eigensolutions of the free vibration of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation. This algorithm first develops the established h-version mesh refinement method for detecting the non-uniform distributed optimised meshes, where the error estimation and element subdivision approaches based on the superconvergent patch recovery displacement method are introduced to obtain high-precision solutions. The errors in the vibration mode solutions in the global space domain are homogenised and approximately the same. Subsequently, on the refined meshes, the algorithm uses higher-order shape functions for the interpolation of trial displacement functions to reduce the errors quickly, until the solution meets a pre-specified error tolerance condition. In this algorithm, the non-uniform mesh generation and higher-order interpolation of shape functions are suitable for addressing the problem of complex frequencies and modes caused by variable structural geometries.FindingsNumerical results are presented for moderately thick circular cylindrical shells with different geometrical factors (circumferential wave number, thickness-to-radius ratio, thickness-to-length ratio) to demonstrate the effectiveness, accuracy and reliability of the proposed method. The hp-version refinement uses fewer optimised meshes than h-version mesh refinement, and only one-step interpolation of the higher-order shape function yields the eigensolutions satisfying the accuracy requirement.Originality/valueThe proposed combination of methodologies provides a complete hp-version adaptive FEM for analysing the free vibration of moderately thick circular cylindrical shells. This algorithm can be extended to general eigenproblems and geometric forms of structures to solve for the frequency and mode quickly and efficiently.

Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Haohan Sun ◽  
Si Yuan
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Mesh Refinement ◽  
Error Tolerance ◽  
Maximum Norm ◽  
Adaptive Finite Element ◽  
Content Type ◽  
Local Mesh Refinement ◽  
Elastic Membranes ◽  
Element Energy Projection

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.

Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

2021 ◽  
Author(s):  
Sundar Namala ◽  
Rizwan Uddin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Solutions ◽  
Second Order ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Integral Methods ◽  
Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

An enriched formulation of isogeometric analysis applied to the dynamical response of bars and trusses

2020 ◽  
Vol 37 (7) ◽  
pp. 2439-2466
Author(s):  
Mateus Rauen ◽  
Roberto Dalledone Machado ◽  
Marcos Arndt
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Isogeometric Analysis ◽  
Computer Aided Design ◽  
Numerical Solutions ◽  
Dynamical Response ◽  
Generalized Finite Element Method ◽  
Content Type ◽  
High Convergence Rate ◽  
Element Method

Purpose This study aims to present a new hybrid formulation based on non-uniform rational b-splines functions and enrichment strategies applied to free and forced vibration of straight bars and trusses. Design/methodology/approach Based on the idea of enrichment from generalized finite element method (GFEM)/extended finite element method (XFEM), an extended isogeometric formulation (partition of unity isogeometric analysis [PUIGA]) is conceived. By numerical examples the methods are tested and compared with isogeometric analysis, finite element method and GFEM in terms of convergence, error spectrum, conditioning and adaptivity capacity. Findings The results show a high convergence rate and accuracy for PUIGA and the advantage of input enrichment functions and material parameters on parametric space. Originality/value The enrichment strategies demonstrated considerable improvements in numerical solutions. The applications of computer-aided design mapped enrichments applied to structural dynamics are not known in the literature.

Stiffness vs flexibility based triangular spring cell – analysis of performance

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
I. St. Doltsinis
Keyword(s):  
Finite Element ◽  
Cell Model ◽  
Rayleigh Quotient ◽  
Cell Models ◽  
Triangular Finite Element ◽  
Content Type ◽  
Triangular Elements ◽  
The Difference ◽  
Analysis Of Performance ◽  
Practical Implications

Purpose The purpose of the present study is to explore the incomplete substitution of the simplex triangular finite element by either of two models: one evolving out as part of the element flexibility, and the other as part of the element stiffness. Design/methodology/approach The elastic energy stored in each of the units under stress or strain decides on stiffer and weaker responses. The pertaining Rayleigh quotient in terms of the flexibility matrices allows bounding the distance of the spring cell models to the finite element in dependence of the triangle configuration. Findings Despite a superiority of the flexibility cell concept observed in computations, the study reveals constellations of shape and stressing of the triangle that favour the stiffness concept. The latter is seen to behave stiffer than its flexibility counterpart and produces results more distant to the finite element in most cases. Research limitations/implications The difference between the stiffness and the flexibility approach to spring cells is investigated for triangular elements in dependence of the geometrical configuration under specific conditions of stressing. This suffices to refute an exclusive superiority of the flexibility concept although largely true. Practical implications The results of the investigation appear useful in deciding between the spring cell models depending on the case of a spring lattice application. Originality/value The flexibility approach to the spring cell is not widely known yet. This cell model deserves a study on performance and comparison to the different, more common stiffness cell model.

scholarly journals A hybrid finite volume/finite element method for shallow water waves by static deformation on seabeds

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Alia Al-Ghosoun ◽  
Ashraf S. Osman ◽  
Mohammed Seaid
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Surface ◽  
Shallow Water ◽  
Finite Volume ◽  
Second Order ◽  
Static Deformation ◽  
Wave Runup ◽  
Content Type ◽  
Element Method

Purpose The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear one-dimensional shallow water equations, including friction terms for the water free-surface and the two-dimensional second-order solid elastostatic equations for the bed deformation. Second, to develop a robust hybrid finite element/finite volume method for solving free-surface runup flow problems over deformable beds. The authors combine the finite volume for free-surface flows and the finite element method for bed elasticity. Design/methodology/approach The authors propose a new model for wave runup by static deformation on seabeds. The model consists of the depth-averaged shallow water system for the water free-surface coupled to the second-order elastostatic formulation for the bed deformation. At the interface between the water flow and the seabed, transfer conditions are implemented. Here, hydrostatic pressure and friction forces are considered for the elastostatic equations, whereas bathymetric forces are accounted for in the shallow water equations. As numerical solvers, the authors propose a well-balanced finite volume method for the flow system and a stabilized finite element method for elastostatics. Findings The developed coupled depth-averaged shallow water system and second-order solid elastostatic system is well suited for modeling wave runup by deformation on seabeds. The derived coupling conditions at the interface between the water flow and the bed topography resolve well the condition transfer between the two systems. The proposed hybrid finite volume element method is accurate and efficient for this class of models. The novel technique used for wet/dry treatment accurately captures the moving fronts in the computational domain without generating nonphysical oscillations. The presented numerical results demonstrate the high performance of the proposed methods. Originality/value Enhancing modeling and computations for wave runup problems is at an early stage in the literature, and it is a new and exciting area of research. To the best of our knowledge, solving wave runup problems by static deformation on seabeds using a hybrid finite volume element method is presented for the first time. The results of this research study, and the research methodologies, will have an important influence on a range of other scientists carrying out research in related fields.

A fictitious crack XFEM with two new solution algorithms for cohesive crack growth modeling in concrete structures

2015 ◽  
Vol 32 (2) ◽  
pp. 473-497 ◽  
Author(s):  
Xiaodong Zhang ◽  
Tinh Quoc Bui
Keyword(s):  
Finite Element ◽  
Nonlinear System ◽  
Crack Growth ◽  
Concrete Structures ◽  
Research Paper ◽  
Cohesive Crack ◽  
Multiple Cracks ◽  
System Of Equations ◽  
Nonlinear System Of Equations ◽  
Content Type

Purpose – The purpose of this paper is to achieve numerical simulation of cohesive crack growth in concrete structures. Design/methodology/approach – The extended finite element method (XFEM) using four-node quadrilateral element associated with the fictitious cohesive crack model is used. A mixed-mode traction-separation law is assumed for the cohesive crack in the fracture process zone (FPZ). Enrichments are considered for both partly and fully cracked elements, and it thus makes the evolution of crack to any location inside the element possible. In all. two new solution procedures based on Newton-Raphson method, which differ from the approach suggested by Zi and Belytschko (2003), are presented to solve the nonlinear system of equations. The present formulation results in a symmetric tangent matrix, conveniently in finite element implementation and programming. Findings – The inconvenience in solving the inversion of an unsymmetrical Jacobian matrix encountered in the existing approach is avoided. Numerical results evidently confirm the accuracy of the proposed approach. It is concluded that the developed XFEM approach is especially suitable in simulating cohesive crack growth in concrete structures. Research limitations/implications – Multiple cracks and crack growth in reinforced concretes should be considered in further studies. Practical implications – The research paper presents a very useful and accurate numerical method for engineering application problems that has ability to numerically simulate the cohesive crack growth of concrete structures. Originality/value – The research paper provides a new numerical approach using two new solution procedures in solving nonlinear system of equations for cohesive crack growth in concrete structures that is very convenient in programming and implementation.

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