Accuracy Of Finite Element Solutions Research Articles

Shape optimization of embedded solids using implicit Vertex-Morphing

One of the biggest challenges in optimizing the shape of complex solids is the requirement to maintain a reasonable mesh quality not only at the boundary but also for the bulk discretization of the interior. Thus, additional regularization and, in many cases, re-meshing of the structure during the iterative process is unavoidable with a Lagrangian description. By tracking the shape update using an Eulerian representation, embedded boundary methods are a promising technique for eliminating mesh distortion problems.This work consistently combines the unique features of implicit Vertex-Morphing and embedded boundary methods, facilitating the node-based shape optimization of solids with industrial complexity. One of the crucial elements for solving the primal problem on a fixed background grid is an efficient and robust quadrature scheme. To this end, we incorporate the open-source C++ framework QuESo (https://github.com/manuelmessmer/QuESo) developed for the numerical integration of arbitrarily complex embedded solids defined by oriented boundary meshes, e.g., in STereoLithography (STL) format. Meanwhile, applying the Helmholtz/Sobolev-based (implicit) filter to the vertices of the embedded boundary mesh not only exploits the extensive design space of node-based optimization but also ensures robust control over the feature size. To realize the above methodology, this work introduces a novel sensitivity analysis that yields mesh-independent shape gradients with respect to the immersed boundary. Through a specific sensitivity weighting, we recover a continuous gradient field from discrete values calculated at the nodes of the background grid. In addition, the presented workflow ensures robust enforcement of challenging geometrical constraints, including minimum wall thicknesses and design space limitations.We critically assess our approach with benchmarks and structures of industrial relevance. In all examples, trivariate B-Spline bases span the background grid, providing highly accurate finite element solutions and shape sensitivities at every iteration. Moreover, the elimination of mesh distortion problems enables a successful termination at the local optimum, even for large shape modifications.

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

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.

FINITE ELEMENT ANALYSIS OF THE DIFFUSION MODEL OF THE BIOCLOGGING OF THE GEOBARRIER

The distribution of an organic chemical and the filtration process in the soil which contains a thin geochemical barrier are considered. Microorganism colonies develop in the presence of organic chemicals in the soil which leads to the so-called phenomenon of bioclogging of the pore space. As a result, the conductivity characteristics of both the soil as a whole and the geochemical barrier change. Conjugation conditions as a component of the mathematical model of chemical filtration in the case of inhomogeneity of porous media and the presence of fine inclusions were modified for the case of bioclogging. The numerical solution of the corresponding nonlinear boundary value problem with modified conjugation conditions was found by the finite element method. The conditions of the existence of a generalized solution of the corresponding boundary value problem are indicated. The results on the theoretical accuracy of finite element solutions are presented. Differences in the value of pressure jumps at a thin geochemical barrier were analyzed for the case considered in the article and the classical case on a model example of filtration consolidation of the soil in the base of solid waste storage. The excess pressure in 600 days after the start of the process reaches 25 % of the initial value when taking into account the effect of bioclogging, while is only 6 % for the test case disregarding the specified effect.

Design Equations for Mixed-Mode Fracture of Dental Ceramic-Cement Interfaces Using the Brazil-Nut-Sandwich Test.

Dental interfaces are subject to mixed-mode loading. This study provides practical guidance for determining interfacial fracture toughness of dental ceramic systems. We address interfacial fracture of a composite resin cement sandwiched between two dental ceramic materials. Emphasis is placed on sandwich disc specimens with cracks originating from elliptical-shaped flaws near the center, for which analytical fracture mechanics methods fail to predict. The interaction integral method is used to provide accurate finite element solutions for cracks with elliptical-shaped flaws in a Brazil-nut-sandwich specimen. The developed model was first validated with existing experimental data and then used to evaluate the three most widely used dental ceramic systems: polycrystalline ceramics (zirconia), glass-ceramics (lithium disilicate), and feldspathic ceramics (porcelain). Contrary to disc specimens with ideal cracks, those with cracks emanating from elliptical-shaped flaws do not exhibit a monotonic increase in interfacial toughness. Also, interfacial fracture toughness is seen to have a direct relationship with the aspect ratio of elliptical-shaped flaws and an inverse relationship with the modulus ratio of the constituents. The presence of an elliptical-shaped flaw significantly changes the interfacial fracture behavior of sandwich structures. Semi-empirical design equations are provided for fracture toughness and stress intensity factors for interfacial cracks. The developed design equations provide practical guidance for determining interfacial fracture toughness of selected dental ceramic material systems. Those equations take into account four critical factors: size of the elliptical flaw, modulus ratio of constituent materials, loading angle, and applied load.

Towards improving finite element solutions automatically with enriched 2D solid elements

In this paper, we propose an automatic procedure to improve the accuracy of finite element solutions using enriched 2D solid finite elements (4-node quadrilateral and 3-node triangular elements). The enriched elements can improve solution accuracy without mesh refinement by adding cover functions to the displacement interpolation of the standard elements. The enrichment scheme is more effective when used adaptively for areas with insufficient accuracy rather than the entire model. For given meshes, an error for each node is estimated, and then proper degrees of cover functions are applied to the selected nodes. A new error estimation method and cover function selection scheme are devised for the proposed adaptive enrichment scheme. Herein, we demonstrate the proposed enrichment scheme through several 2D problems.

A novel four-node inverse-plate element for shape and stress sensing of laminated composite and sandwich plates

Kompozit malzemeler, çeşitli mühendislik yapılarının birincil yük taşıyıcı bileşenlerinin imalat sürecinde yaygın olarak kullanılmaktadır. Ancak, kompozitlerin yapısal bütünlüğünün, yapının hizmet ömrü boyunca meydana gelen hasarlardan dolayı azalması muhtemeldir. Duruma dayalı bakım programlaması için kompozit yapılara gerçek zamanlı şekil ve gerilme algılaması gerçekleştirebilen bir yapısal sağlık izleme sistemi entegre edilebilir. Bu çalışmada, Rafine Zikzak Teorisinin (RZT) temel olarak kullanıldığı ters sonlu elemanlar yöntemine (iFEM) dayanan yeni bir dört düğümlü ters plaka elemanı önerilmiştir. Buradaki iFEM-RZT formülasyonu, membran, eğilme, enine kayma ve zikzak kesit gerinimlerini içeren RZT'nin bütün gerinim ölçüleri kümesini kullanan ağırlıklı en küçük kareler fonksiyonunu en aza indirgemektedir. Mevcut elemanın temel faydası, herhangi bir yükleme bilgisi gerektirmemesi ve şekil algılaması için yalnızca yerleşik sensörlerden alınan gerinim ölçer ölçümlerini kullanmasıdır. Diğer bir avantajı, aynı zamanda genel bir lamine kompozit ve sandviç yapı sınıfının tam alan ve üç boyutlu yer değiştirmelerini ve gerilmelerini tahmin etmek içinde uygulanabilir olmasıdır. Mevcut elemanın potansiyel yeteneklerini göstermek için, farklı laminasyon tiplerine sahip olan bir lamine kompozit plaka üzerinde çeşitli vaka çalışmaları yapılmıştır. İFEM-RZT sonuçları ve son derece hassas sonlu elemanlar çözümleri arasındaki karşılaştırmaya göre, mevcut yaklaşımın üstün doğruluğu ortaya konmuştur.

Modeling the non-trivial behavior of anisotropic beams: A simple Timoshenko beam with enhanced stress recovery and constitutive relations

In this work we analyze the non-trivial influence of material anisotropy on the structural behavior of a multi-layer planar beam. Indeed, analytical results available in literature are limited to homogeneous beams and several aspects have not been addressed yet, preventing a deep understanding of the mechanical response of anisotropic structural elements. This paper proposes an effective recovery of stress distributions and an energetically consistent evaluation of constitutive relations to be used within a planar Timoshenko beam model. The resulting structural analysis tool highlights the following peculiarities of anisotropic beams: (i) the transversal internal force affects the maximum axial stress up to 30%, and (ii) the anisotropy influences the beam displacements more than the standard shear deformation, even for extremely slender beams. A rigorous comparison with analytical and accurate 2D Finite Element solutions confirms the accuracy of the proposed approach, which leads to errors usually below 2%.

Numerical frequency analysis of skew sandwich layered composite shell structures under thermal environment including shear deformation effects

The numerical thermal frequency responses of the skew sandwich shell panels structure are investigated via a higher-order polynomial shear deformation theory including the thickness stretching effect. A customized MATLAB code is developed using the current mathematical model for the computational purpose. The finite element solution accuracy and consistency have been checked via solving different kinds of numerical benchmark examples taken from the literature. After confirming the standardization of the model, it is further extended to show the effect of different important geometrical parameters such as span-to-thickness ratios, aspect ratios, curvature ratios, core-to-face thickness ratios, skew angles, and support conditions on the frequencies of the sandwich composite flat/curved panel structure under elevated temperature environment.

Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.

Anisotropic mesh generation methods with variable metrics and natural metric for anisotropic elliptic equation

A suitable anisotropic mesh is important for the anisotropic problem in many applications. Given a suitable metric tensor and using an efficient algorithm to generate the anisotropic mesh are crucial steps in the anisotropic mesh methodology. In this paper, we develop a new anisotropic mesh generation method by combining the modified anisotropic Delaunay criterion and force-based equilibrium smoothing function on an anisotropic background mesh. We observe the natural metric for the anisotropic problem with a variable anisotropic diffusion is the inverse of the diffusion in three beneficial aspects: Better discrete algebraic systems with smaller condition numbers, more accurate finite element solution and superconvergence in $l^2$ norm on the mesh nodes. Various numerical examples are presented to demonstrate the effectiveness of the proposed method and natural metric.

Anisotropic mesh generation methods based on ACVT and natural metric for anisotropic elliptic equation

Anisotropic meshes are known to be well-suited for problems which exhibit anisotropic solution features. Defining an appropriate metric tensor and designing an efficient algorithm for anisotropic mesh generation are two important aspects of the anisotropic mesh methodology. In this paper, we are concerned with the natural metric tensor for use in anisotropic mesh generation for anisotropic elliptic problems. We provide an algorithm to generate anisotropic meshes under the given metric tensor. We show that the inverse of the anisotropic diffusion matrix of the anisotropic elliptic problem is a natural metric tensor for the anisotropic mesh generation in three aspects: better discrete algebraic systems, more accurate finite element solution and superconvergence on the mesh nodes. Various numerical examples demonstrating the effectiveness are presented.

Evolutionary topology optimization using the extended finite element method and isolines

This study presents a new algorithm for structural topological optimization of two-dimensional continuum structures by combining the extended finite element method (X-FEM) with an evolutionary optimization algorithm. Taking advantage of an isoline design approach for boundary representation in a fixed grid domain, X-FEM can be implemented to improve the accuracy of finite element solutions on the boundary during the optimization process. Although this approach does not use any remeshing or moving mesh algorithms, final topologies have smooth and clearly defined boundaries which need no further interpretation. Numerical comparisons of the converged solutions with standard bi-directional evolutionary structural optimization solutions show the efficiency of the proposed method, and comparison with the converged solutions using MSC NASTRAN confirms the high accuracy of this method.

Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation

Mesh quality plays an essential role in finite element applications, since it affects the efficiency of the simulation with respect to solution accuracy and computational effort. Therefore, mesh smoothing techniques are often applied for improving mesh quality while preserving mesh topology. One of these methods is the recently proposed geometric element transformation method (GETMe), which is based on regularizing element transformations. It will be shown numerically that this smoothing method is particularly suitable, from an applicational point of view, since it leads to a significant reduction of discretization errors within the first few smoothing steps requiring only little computational effort. Furthermore, due to reduced condition numbers of the stiffness matrices the performance of iterative solvers of the resulting finite element equations is improved. This is demonstrated for the Poisson equation with a number of meshes of different complexity and type as well as for piecewise linear and quadratic finite element basis functions. Results are compared to those obtained by two variants of Laplacian smoothing and a state of the art global optimization-based approach.

Serendipity 요소법에 의한 전기비저항 3차원 모델링

전기비저항 탐사는 자동측정과 정밀한 자료 획득이 가능해 지면서 토목 및 환경문제 등 다양한 분야에 적용되고 있다. 이에 따라 시간에 따른 지하의 변화를 정밀하게 파악할 수 있는 전기비저항 모니터링 기법이 도입되면서 시간경과 모니터링 자료의 보다 정확한 모델링 기법과 역산 기법의 개발이 요구된다. 여기서는 3차원 전기비저항 모델링으로 요소의 변형을 통해 임의 형상의 이상체 및 복잡한 지형의 굴곡을 표현하기 쉬운 유한요소법을 사용하였다. 유한요소법에서 선형요소(1차 요소)는 시스템 방정식의 구성이 간단하고 대역폭이 좁다는 장점이 있다. 하지만 선형요소는 요소 또는 절점의 수에 따라 해의 수렴속도가 느리며 또한 정확성에 한계가 있다. 일반적으로 유한요소법에서 해의 정확성을 높이기 위해 고차요소를 사용한다. 본 논문에서는 고차의 Serendipity 요소를 사용하는 3차원 전기비저항 모델링 프로그램을 개발하였다. 선형요소법과 Serendipity 요소법의 비교를 위해 직육면체의 이상체 모델에 적용하였을 때, 선형요소법의 결과에 비해 Serendipity 요소를 사용하는 3차원 전기비저항 모델링의 결과에서 보다 정확하게 나타나는 것을 확인하였다. A resistivity method has been applied to wide range of engineering and environmental problems with the help of automatic and precise data acquisition. Thus, more accurate modeling and inversion of time-lapse monitoring data are required since resistivity monitoring has been introduced to quantitatively find out subsurface changes With respect to time. Here, we used the finite element method (FEM) for 3D resistivity modeling since the method is easy to realize complex topography and arbitrary shaped anomalous bodies. In the FEM, the linear elements, also referred to as first order elements, have certain advantages of simple formulation and narrow bandwidth of system equation. However, the linear elements show the poor accuracy and slow convergence of the solution with respect to the number of elements or nodes. To achieve the higher accuracy of finite element solution, high order elements are generally used. In this study, we developed a 3D resistivity modeling program using high order Serendipity elements. Comparing the Serendipity element solutions for a cube model with the linear element solutions, we assured that the Serendipity element solutions are more accurate than the linear element solutions in the 3D resistivity modeling.

A partition-of-unity based ‘FE-Meshfree’ QUAD4 element with radial-polynomial basis functions for static analyses

Recently, a partition-of-unity (PU) based finite element called ‘FE-Meshfree’ QUAD4 element has been published. That element employed polynomial basis functions for the local approximation (LA). In the present paper, a new FE-Meshfree QUAD4 element employing hybrid radial-polynomial basis functions for the LA is proposed. An advantage of radial-polynomial basis is the freedom from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions. Another advantage is the improved accuracy of finite element solution. The new element has been applied to several linear and nonlinear test problems. The results demonstrate that the new element gives much better performance as compared to the previous FE-Meshfree QUAD4 element with pure polynomial basis. Even with a lower order basis, viz., with just three or four polynomial terms, the present element is capable of giving more accurate solution than the previous element which employs a six-term polynomial basis. The new element also exhibits a much higher degree of tolerance to mesh distortion than the known quadratic elements. Moreover, the linear dependence problem, otherwise associated with many of the partition-of-unity (PU) based elements, is completely eliminated from the present element.

Generalized finite element enrichment functions for discontinuous gradient fields

AbstractA general GFEM/XFEM formulation is presented to solve two‐dimensional problems characterized by C0 continuity with gradient jumps along discrete lines, such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media. The new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines, such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks. We show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines. The use of the proposed enrichments in both linear and quadratic approximations is investigated, as well as their combination with interface enrichment functions available in the literature. Through a detailed convergence study, we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems. In the linear case, the effectiveness of correction functions proposed in the literature is also investigated. Copyright © 2009 John Wiley & Sons, Ltd.

Torsion of functionally graded hollow tubes

An analytical formulation is presented for torsional analysis of functionally graded hollow tubes of arbitrary shape. Moreover, relatively simpler formulas are presented for torsional analysis of functionally graded hollow tubes of polygonal shape. Thicknesses of all segments of the cross section are the same. Shear modulus of rigidity changes continuously across the thickness between two phases according to a power law distribution. Governing equations in terms of Prandtl's stress function are used to derive the formulas. Several examples are presented to show the accuracy and efficiency of the formulation. The obtained results are verified by accurate finite element solutions. Effects of changing thickness and volume fraction of constituent materials are also investigated. Derived formulas can be useful for analysis of thin to moderately thick-walled hollow tubes.

Finite element analysis of rough surfaces based on ultimate‐stress asperity concept

AbstractAn improved mathematical elastic–plastic model for the contact of rough surfaces that is based on an accurate finite element solution of a deformable single asperity and a rigid flat surface is developed to provide dimensionless expressions for the contact area and contact load. This model differs from the existing models, in that it accounts for the level of interference beyond expected failure. The finite element solution is used to define the limits at which failure occurs. A fictitious ultimate‐stress asperity is introduced to facilitate the derivation of the contact model. This model considers a realistic picture of elastic–plastic deformation where elastic, plastic and failure behaviors can occur simultaneously for an asperity. Subsequent comparison of the results for estimating contact area and load using the present model and the earlier methods shows identical results for pure elastic contacts with plasticity index values at about 0.5 but substantial difference for the net elastic–plastic contacts having plasticity index values above 0.8. When plasticity index reaches 6 and beyond, the three models predict similar total contact area and load values and that the contact is purely plastic. Copyright © 2008 John Wiley & Sons, Ltd.

Torsion of moderately thick hollow tubes with polygonal shapes

A simple formulation is presented for torsion analysis of hollow tubes with polygonal shapes. Thicknesses of segments of cross section can be different. Governing equations in term of Prandtl’s stress function are used to derive the formulas. The derived formulas are so simple that computations can be carried out with a pocket calculator. Several examples are presented to show the accuracy and efficiency of the formulation. The obtained results are verified by accurate finite element solutions. It will be seen that the derived formulas can be useful for analysis of thin-walled and moderately thick-walled hollow tubes.

The h-, p-, and r-refinements of Finite Element Analysis of Flexible Slider Crank Mechanism

Three common refinement methods of achieving more accurate finite element solutions are to increase the number of elements, to employ higher-degree interpolation functions, and to implement an adaptive mesh by moving the nodes but maintaining the same number of elements as well as the degree of interpolation functions. This paper presents a finite element analysis by applying these refinements to a flexible slider crank mechanism. The formulation is based on the Euler—Lagrange equation, for which the Lagrangian includes the components related to the kinetic energy, the strain energy, and the work done by axial loads in a link that undergoes elastic transverse deflection. A beam element is modeled based on a translating and rotating motion. An error analysis is demonstrated by defining several error indicators, which are based on energy of an entire mechanism, transverse displacement, and bending strain of midpoints on crank and coupler. This paper also demonstrates the effect of the stiffness of the crank.