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

2021

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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.

Section: Example 1: H-version and Hp-version Adaptive Finite Element ...mentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

“…Example 1: errors in the computed frequencies and corresponding final meshes, between the h - (Wang et al , 2018; Wang, 2020) and hp -version adaptive finite element algorithms…”

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Section: Numerical Examplesmentioning

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Section: Introductionmentioning

confidence: 99%

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

2021

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Stress-Dependent Unstable Dynamic Propagation of Three-Dimensional Multiple Hydraulic Fractures with Improved Fracturing Sequences in Heterogeneous Reservoirs: Numerical Cases Study via Poroelastic Effective Medium Model

Liu

2021

Energy Fuels

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Multistage hydrofracturing of horizontal wells is one of the key technologies for deep tight oil and gas resource exploitation. The stable and parallel propagation of multiple fractures formed by multistage perforation cluster fracturing is conducive to the formation of a developed fracture network. However, in the actual process of rock fracturing in deep tight reservoirs, owing to the influence of many factors, such as fracturing sequence, perforation cluster space, reservoir heterogeneity, and stress shadow effect, the three-dimensional (3D) fracture network that propagates unsteadily of multiple fractures is often formed. This has become an important problem limiting artificial transformation and in optimizing fracture networks. In view of the above difficulties, a 3D finite element–discrete element model considering fluid–solid coupling and fracturing fluid leak-off effect is proposed to simulate hydrofracturing, and an effective medium model to describe heterogeneous reservoir rock mass is established. Several typical fracturing sequences (sequential, alternate, simultaneous, and two improved alternate scenarios), perforation cluster spaces (100, 75, 50, and 25 m), and heterogeneous reservoir conditions were analyzed. The spatial unstable propagation, stress field interference, and evolution behavior of the pressure fracture network were simulated, and the quantitative results of the dynamic propagation shape, fracture propagation area, and spatial volume of the pressure fracture network were obtained. The numerical results show that when the perforation cluster space decreases, the stress shadow effect between hydraulic fractures intensifies, and the 3D fracture exhibits unstable propagation and spatial deflection. Compared with sequential and simultaneous fracturing, all types of improved alternate fracturing can effectively reduce the stress interference between fractures, fracture deflection, and form the developed fracture network area. Compared with the homogeneous reservoir model, the heterogeneity of the reservoir rock mass becomes an important factor that limits the propagation of the fracture network. These results can provide a basis for understanding the mechanisms of stress-dependent unstable dynamic propagation of 3D multiple hydraulic fractures of improved fracturing sequences in heterogeneous reservoirs.

Finite element mesh refinement for in-plane and out-of-plane vibration of variable geometrical Timoshenko beams based on superconvergent vibration modes

2022

PurposeIn this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.Design/methodology/approachAn adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.FindingsThe results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.Originality/valueThe proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.