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Displacement based polytopal elements a strain smoothing and scaled boundary approach Bordas, Stéphane ; Scientific Conference (2019, May 03) Detailed reference viewed: 64 (4 UL)ADVANCES IN GEOMETRY INDEPENDENT APPROXIMATIONS ; ; Bordas, Stéphane et al Scientific Conference (2019, April 11) We present recent advances in geometry independent field approximations. The GIFT approach is a generalisation of isogeometric analysis where the approximation used to describe the field variables no ... [more ▼] We present recent advances in geometry independent field approximations. The GIFT approach is a generalisation of isogeometric analysis where the approximation used to describe the field variables no-longer has to be identical to the approximation used to describe the geometry of the domain. As such, the geometry can be described using usual CAD representations, e.g. NURBS, which are the most common in the CAD area, whilst local refinement and meshes approximations can be used to describe the field variables, enabling local adaptivity. We show in which cases the approach passes the patch test and present applications to various mechanics, fracture and multi-physics problems. [less ▲] Detailed reference viewed: 137 (6 UL)A volume-averaged nodal projection method for the Reissner-Mindlin plate model ; ; Hale, Jack et al in Computer Methods in Applied Mechanics & Engineering (2018), 341 We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and ... [more ▼] We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and is devoid of shear-locking. The proposed approach uses linear maximum-entropy approximations and is built variationally on a two-field potential energy functional wherein the shear strain, written in terms of the primitive variables, is computed via a volume-averaged nodal projection operator that is constructed from the Kirchhoff constraint of the three-field mixed weak form. The stability of the method is rendered by adding bubble-like enrichment to the rotation degrees of freedom. Some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses. [less ▲] Detailed reference viewed: 97 (13 UL)Isogeometric analysis of thin Reissner-Mindlin plates and shells: Locking phenomena and generalized local B-bar method Hu, Qingyuan ; ; et al E-print/Working paper (2017) We propose a generalized local $\bar{B}$ framework, addressing locking in degenerated Reissner-Mindlin plate and shell formulations in the context of isogeometric analysis. Parasitic strain components are ... [more ▼] We propose a generalized local $\bar{B}$ framework, addressing locking in degenerated Reissner-Mindlin plate and shell formulations in the context of isogeometric analysis. Parasitic strain components are projected onto the physical space locally, i.e. at the element level, using a least-squares approach. The formulation is general and allows the flexible utilization of basis functions of different order as the projection bases. The present formulation is much cheaper computationally than the global $\bar{B}$ method. Through numerical examples, we show the consistency of the scheme, although the method is not Hu-Washizu variationally consistent. The numerical examples show that the proposed formulation alleviates locking and yields good accuracy for various thicknesses, even for slenderness ratios of $1 \times 10^5$, and has the ability to capture deformations of thin shells using relatively coarse meshes. From the detailed numerical study, it can be opined that the proposed method is less sensitive to locking and mesh distortion. [less ▲] Detailed reference viewed: 201 (14 UL)Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties ; ; et al in Asia Pacific Journal on Computational Engineering (2017) In this paper, the accuracy and the convergence properties of Trefftz finite element method over arbitrary polygons are studied. Within this approach, the unknown displacement field within the polygon is ... [more ▼] In this paper, the accuracy and the convergence properties of Trefftz finite element method over arbitrary polygons are studied. Within this approach, the unknown displacement field within the polygon is represented by the homogeneous solution to the governing differential equations, also called as the T-complete set. While on the boundary of the polygon, a conforming displacement field is independently defined to enforce the continuity of the field variables across the element boundary. An optimal number of T-complete functions are chosen based on the number of nodes of the polygon and the degrees of freedom per node. The stiffness matrix is computed by the hybrid formulation with auxiliary displacement frame. Results from the numerical studies presented for a few benchmark problems in the context of linear elasticity show that the proposed method yields highly accurate results with optimal convergence rates. [less ▲] Detailed reference viewed: 26 (1 UL)A linear smoothed quadratic finite element for the analysis of laminated composite Reissner–Mindlin plates ; ; et al in Composite Structures (2017), 180 It is well known that the high-order elements have significantly improved the accuracy of solutions in the traditional finite element analysis, but the performance of high-order elements is restricted by ... [more ▼] It is well known that the high-order elements have significantly improved the accuracy of solutions in the traditional finite element analysis, but the performance of high-order elements is restricted by the shear-locking and distorted meshes for the plate problems. In this paper, a linear smoothed eight-node Reissner-Mindlin plate element (Q8 plate element) based on the first order shear deformation theory is developed for the static and free vibration analysis of laminated composite plates, the computation of the interior derivatives of shape function and isoparametric mapping can be removed. The strain matrices are modified with a linear smoothing technique by using the divergence theorem between the nodal shape functions and their derivatives in Taylor’s expansion. Moreover, the first order Taylor’s expansion is also employed for the construction of stiffness matrix to satisfy the linear strain distribution. Several numerical examples indicate that the novel Q8 plate element has good performance to alleviate the shear-locking phenomenon and improve the quality of the solutions with distorted meshes. [less ▲] Detailed reference viewed: 41 (2 UL)A fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities ; ; et al in International Journal for Numerical Methods in Engineering (2017), 110(3), 203-226 In this paper, we propose a fully smoothed extended finite element method (SmXFEM) for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms ... [more ▼] In this paper, we propose a fully smoothed extended finite element method (SmXFEM) for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms in the stiffness and mass matrixes can be computed by smoothing technique. This is accomplished by combining the Green’s divergence theorem with the evaluation of indefinite integral based on smoothing technique, which is used to transform the domain integral into boundary integral. The proposed technique completely eliminates the need for isoparametric mapping and the computing of Jacobian matrix even for the mass matrix. When employed over the enriched elements, the proposed technique does not require sub-triangulation for the purpose of numerical integration. The accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities. It can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion. [less ▲] Detailed reference viewed: 38 (2 UL)A linear smoothed higher-order CS-FEM for the analysis of notched laminated composites ; ; et al in Engineering Analysis with Boundary Elements (2017), 85 Higher-order elements with highly accurate solutions are attractive for stress analysis and stress concentration problems. However, the distorted eight-node serendipity quadrilateral element is known to ... [more ▼] Higher-order elements with highly accurate solutions are attractive for stress analysis and stress concentration problems. However, the distorted eight-node serendipity quadrilateral element is known to yield inaccurate re- sults and sub-optimal convergence rate. In this paper, we present a higher order CS-FEM to alleviate the effect of distorted mesh and guarantee the quality of solutions by employing a linear smoothing technique over eight-node quadratic serendipity elements. The modified. strain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor’s expansion of the weak form. The proposed method eliminates the need for isoparametric mapping and numerical studies demonstrate that the proposed method is insensitive to mesh distortion. The improved accuracy and superior convergence rates are numerically demon- strated with a few benchmark problems. The analysis of the stress concentration around cutouts also proves that the present method has good performance for the laminated composites. [less ▲] Detailed reference viewed: 51 (1 UL)Numerical methods for fracture/cutting of heterogeneous materials Sutula, Danas ; Agathos, Konstantinos ; Ziaei Rad, Vahid et al Presentation (2016, December) Detailed reference viewed: 167 (15 UL)Linear smoothing over arbitrary polytopes for compressible and nearly incompressible linear elasticity ; Tomar, Satyendra ; Bordas, Stéphane et al Scientific Conference (2016, June) We present a displacement based approach over arbitrary polytopes for compressible and nearly incompressible linear elastic solids. In this approach, a volume-averaged nodal projection operator is ... [more ▼] We present a displacement based approach over arbitrary polytopes for compressible and nearly incompressible linear elastic solids. In this approach, a volume-averaged nodal projection operator is constructed to project the dilatational strain into an approximation space of equal or lower-order than the approximation space for the displacement field, resulting in a locking-free method. The formulation uses the usual Wachspress interpolants over arbitrary polytopes and the stability of the method is ensured by the addition of bubble like functions. The smoothed strains are evaluated based on the linear smoothing procedure. This further softens the bilinear form allowing the procedure to search for a solution satisfying the divergence- free condition. The divergence-free condition of the proposed approach is verified through systematic numerical study. The formulation delivers optimal convergence rates in the energy and L2-norms. Inf-sup tests are presented to demonstrated the stability of the formulation. [less ▲] Detailed reference viewed: 185 (3 UL)Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods ; Bordas, Stéphane ; in International Journal for Numerical Methods in Engineering (2015), 104(13), 1173-1199 We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally ... [more ▼] We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM.We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub-triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the O.dof s/1:1 in case of the conventional polygonal FEM, while it scales as O.dof s/0:7 in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. [less ▲] Detailed reference viewed: 48 (1 UL)A hybrid T-Trefftz polygonal finite element for linear elasticity ; ; Bordas, Stéphane E-print/Working paper (2014) In this paper, we construct hybrid T-Trefftz polygonal finite elements. The displacement field within the polygon is repre- sented by the homogeneous solution to the governing differential equation, also ... [more ▼] In this paper, we construct hybrid T-Trefftz polygonal finite elements. The displacement field within the polygon is repre- sented by the homogeneous solution to the governing differential equation, also called as the T-complete set. On the boundary of the polygon, a conforming displacement field is independently defined to enforce continuity of the displacements across the element boundary. An optimal number of T-complete functions are chosen based on the number of nodes of the polygon and degrees of freedom per node. The stiffness matrix is computed by the hybrid formulation with auxiliary displacement frame. Results from the numerical studies presented for a few benchmark problems in the context of linear elasticity shows that the proposed method yield highly accurate results. [less ▲] Detailed reference viewed: 92 (3 UL)On the equivalence between the cell-based smoothed finite element method and the virtual element method ; Bordas, Stéphane ; E-print/Working paper (2014) We revisit the cell-based smoothed finite element method (SFEM) for quadrilateral elements and extend it to arbitrary polygons and polyhedrons in 2D and 3D, respectively. We highlight the similarity ... [more ▼] We revisit the cell-based smoothed finite element method (SFEM) for quadrilateral elements and extend it to arbitrary polygons and polyhedrons in 2D and 3D, respectively. We highlight the similarity between the SFEM and the virtual element method (VEM). Based on the VEM, we propose a new stabilization approach to the SFEM when applied to arbitrary polygons and polyhedrons. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Later, the SFEMis combined with the scaled boundary finite element method to problems involving singularity within the framework of the linear elastic fracture mechanics in 2D. [less ▲] Detailed reference viewed: 381 (10 UL)smooth nodal stress in the XFEM for crack propagation simulations ; Bordas, Stéphane ; Scientific Conference (2013, September) In this paper, we present a method to achive smooth nodal stresses in the XFEM without post-processing. This method was developed by borrowing ideas from ``twice interpolating approximations'' (TFEM) [1 ... [more ▼] In this paper, we present a method to achive smooth nodal stresses in the XFEM without post-processing. This method was developed by borrowing ideas from ``twice interpolating approximations'' (TFEM) [1]. The salient feature of the method is to introduce an ``average'' gradient into the construction of the approximation, resulting in improved solution accuracy, both in the vicinity of the crack tip and in the far field. Due to the high order polynomial basis provided by the interpolants, the new approximation enhances the smoothness of the solution without requiring an increased number of degrees of freedom. This is particularly advantageous for low-order elements and in fracture mechanics, where smooth stresses are important for certain crack propagation criteria, e.g. based on maximum principal stresses. Since the new approach adopts the same mesh discretization, i.e. simplex meshes, it can be easily extended into various problems and is easily implemented. We discuss the increase in the bandwidth which is the major drawback of the present method and can be somewhat alleviated by using an element-by-element solution strategy. Numerical tests show that the new method is as robust as XFEM, considering precision, model size and post-processing time. By comparing in detail the behaviour of the method on crack propagation examples, we can conclude that for two-dimensional problems, the proposed method tends to be an efficient alternative to the classical XFEM [2][3] especially when local, stress-based propagation criteria are used. [less ▲] Detailed reference viewed: 338 (4 UL)Supersonic flutter analysis of functionally graded material plates with cracks ; ; Bordas, Stéphane in Frontiers in Aerospace Engineering (2013), 2(2), 91--97 In this paper, the flutter behaviour of functionally graded material plates immersed in a supersonic flow is studied. An enriched 4-noded quadrilateral element based on field consistency approach is used ... [more ▼] In this paper, the flutter behaviour of functionally graded material plates immersed in a supersonic flow is studied. An enriched 4-noded quadrilateral element based on field consistency approach is used for this study. The crack is modelled independent of the underlying mesh using partition of unity method (PUM), the extended finite element method (XFEM). The material properties are assumed to be graded only in the thickness direction and the effective material properties are estimated using the rule of mixtures. The plate kinematics is described based on the first order shear deformation theory (FSDT) and the shear correction factors are evaluated employing the energy equivalence principle. The influence of the crack length, the crack orientation, the flow angle and the gradient index on the aerodynamic pressure and the frequency are numerically studied. The results obtained here reveal that the critical frequency and pressure decrease with increase in crack the length and are minimum when the crack is aligned to the flow angle. [less ▲] Detailed reference viewed: 380 (4 UL)Numerical integration over arbitrary surfaces in partition of unity finite elements ; ; et al Scientific Conference (2009, September) Detailed reference viewed: 87 (0 UL)On the Smoothed eXtended Finite Element Method for Continuum ; Bordas, Stéphane ; et al Scientific Conference (2009, April) In this paper, we combine the strain smoothing technique proposed by Liu et al [1] coined as the smoothed finite element method (SFEM) to partition of unity methods, namely the extended finite element ... [more ▼] In this paper, we combine the strain smoothing technique proposed by Liu et al [1] coined as the smoothed finite element method (SFEM) to partition of unity methods, namely the extended finite element method (XFEM) [2] to give birth to the smoothed extended finite element method (SmXFEM) [3]. SmXFEM shares properties both with the SFEM and the XFEM. The proposed method eliminates the need to compute and integrate the derivatives of shape functions (which are singular at the tip for linear elastic fracture mechanics). The need for isoparametric mapping is eliminated because the integration is done along the boundary of the finite element or smoothing cells, which allows elements of arbitrary shape. We present numerical results for various differential equations that have singularity or steep gradient at the boundary. The method is verified on several examples and comparisons are made to the conventional XFEM. [less ▲] Detailed reference viewed: 87 (1 UL)A novel numerical integration technique over arbitrary polygons ; ; Bordas, Stéphane et al Scientific Conference (2009, April) In this paper, a new numerical integration technique [1] on arbitrary polygons is presented. The polygonal do- main is mapped conformally to the unit disk using Schwarz-Christoffel mapping [2] and a ... [more ▼] In this paper, a new numerical integration technique [1] on arbitrary polygons is presented. The polygonal do- main is mapped conformally to the unit disk using Schwarz-Christoffel mapping [2] and a midpoint quadrature rule defined on the unit circle is used. This method eliminates the need for a two level isoparametric mapping usuall required [3]. Moreover the positivity of the Jacobian is guaranteed. We present numerical results for a few benchmark problems in the context of polygonal finite elements that show the effectiveness of the method. [less ▲] Detailed reference viewed: 80 (0 UL)Linear smoothing over arbitrary polytopes ; ; et al Scientific Conference (n.d.) The conventional constant strain smoothing technique yields less accurate solutions that other techniques such as the conventional polygonal finite element method [1, 2]. In this work, we propose a linear ... [more ▼] The conventional constant strain smoothing technique yields less accurate solutions that other techniques such as the conventional polygonal finite element method [1, 2]. In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex poly- topes. The method relies on sub-division of the polytope into simplical subcells; however instead of using a constant smoothing function, we employ a linear smoothing function over each subcell. This gives a new definition for the strain to compute the stiffness matrix. The convergence properties and accuracy of the proposed scheme are discussed by solving few benchmark problems. Numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the optimal convergence rate as in traditional quadrilateral and hexahedral finite elements. The accuracy is also improved, and all the methods tested pass the patch test to machine precision. [less ▲] Detailed reference viewed: 106 (0 UL)Linear smoothed extended finite element method ; ; et al E-print/Working paper (n.d.) The extended finite element method (XFEM) was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to ... [more ▼] The extended finite element method (XFEM) was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to be split by a discontinuity, strong or weak and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes. Moreover, in the case of open surfaces and singularities, special, usually non-polynomial functions must also be integrated.A variety of approaches have been proposed to facilitate these special types of numerical integration, which have been shown to have a large impact on the accuracy and convergence of the numerical solution. The smoothed extended finite element method (SmXFEM) [1], for example, makes numerical integration elegant and simple by transforming volume integrals into surface integrals. However, it was reported in [1, 2] that the strain smoothing is inaccurate when non-polynomial functions are in the basis. This is due to the constant smoothing function used over the smoothing domains which destroys the effect of the singularity. In this paper, we investigate the benefits of a recently developed Linear smoothing procedure [3] which provides better approximation to higher order polynomial fields in the basis. Some benchmark problems in the context of linear elastic fracture mechanics (LEFM) are solved to compare the standard XFEM, the constant-smoothed XFEM (Sm-XFEM) and the linear-smoothed XFEM (LSm-XFEM). We observe that the convergence rates of all three methods are the same. The stress intensity factors (SIFs) computed through the proposed LSm-XFEM are however more accurate than that obtained through Sm-XFEM. To conclude, compared to the conventional XFEM, the same order of accuracy is achieved at a relatively low computational effort. [less ▲] Detailed reference viewed: 62 (3 UL) |
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