iMechanica - Mixed finite element methods; finite element exterior calculus; nonlinear elasticity; incompressible elasticity; Hilbert complex
https://imechanica.org/taxonomy/term/11923
enA compatible mixed finite element method for large deformation analysis of solids in spatial configuration
https://imechanica.org/node/26027
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/11923">Mixed finite element methods; finite element exterior calculus; nonlinear elasticity; incompressible elasticity; Hilbert complex</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In this work, a new mixed finite element formulation is presented for the analysis of two-dimensional compressible solids in finite strain regime. A three-field Hu-Washizu functional, with displacement, displacement gradient and stress tensor considered as independent fields, is utilized to develop the formulation in spatial configuration. Certain constraints are imposed on displacement gradient and stress tensor so that they satisfy the required continuity conditions across the boundary of elements. From theoretical standpoint, simplex elements are best suited for the application of continuity constraints. The techniques that are proposed to implement the constraints facilitate their automatic imposition and, hence, they can be regarded as an important feature of the work. Since the exterior calculus provides the basis for the developments presented herein, the relevant topics are discussed within the context of the work. Various technical aspects of the formulation are described in detail. These aspects help to illuminate the mathematical formulation that might seem vague in the first place and, more importantly, they help to provide an efficient implementation for ensuing developments. The performance of the mixed finite element method is studied through benchmark numerical examples and it is compared with other similar elements. It is shown that the element has excellent convergence properties and it is numerically stable, especially for problems where classical first order elements demonstrate stiff or unstable behavior.</p>
<p><a href="https://onlinelibrary.wiley.com/doi/10.1002/nme.6978">https://onlinelibrary.wiley.com/doi/10.1002/nme.6978</a></p>
</div></div></div>Thu, 09 Jun 2022 00:12:09 +0000M. Jahanshahi26027 at https://imechanica.orghttps://imechanica.org/node/26027#commentshttps://imechanica.org/crss/node/26027Compatible-Strain Mixed Finite Element Methods for 3D Compressible and Incompressible Nonlinear Elasticity
https://imechanica.org/node/23532
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/11923">Mixed finite element methods; finite element exterior calculus; nonlinear elasticity; incompressible elasticity; Hilbert complex</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>A new family of mixed finite element methods --- <em>compatible-strain mixed finite element methods</em> (CSFEMs) --- are introduced for three-dimensional compressible and incompressible nonlinear elasticity. A Hu-Washizu-type functional is extremized in order to obtain a mixed formulation for nonlinear elasticity. The independent fields of the mixed formulations are the displacement, the displacement gradient, and the first Piola-Kirchhoff stress. A pressure-like field is also introduced in the case of incompressible elasticity. We define the displacement in H^1, the displacement gradient in H(curl), the stress in H(div), and the pressure-like field in L^2. In this setting, for improving the stability of the proposed finite element methods without compromising their consistency, we consider some stabilizing terms in the Hu-Washizu-type functional that vanish at its critical points. Using a conforming interpolation, the solution and the test spaces are approximated with some piecewise polynomial subspaces of them. In three dimensions, this requires using the Nedelec edge elements for the displacement gradient and the Nedelec face elements for the stress. This approach results in mixed finite element methods that satisfy the Hadamard jump condition and the continuity of traction on all the internal faces of the mesh. This, in particular, makes CSFEMs quite efficient for modeling heterogeneous solids. We assess the performance of CSFEMs by solving several numerical examples, and demonstrate their good performance for bending problems, for bodies with complex geometries, and in the near-incompressible and the incompressible regimes. Using CSFEMs, one can capture very large strains and accurately approximate stresses and the pressure field. Moreover, in our numerical examples, we do not observe any numerical artifacts such as checkerboarding of pressure, hourglass instability, or locking. </p>
</div></div></div><div class="field field-name-upload field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><table class="sticky-enabled">
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://imechanica.org/files/3DMFEMIncmpNLElasFaYa2019.pdf" type="application/pdf; length=1495268">3DMFEMIncmpNLElasFaYa2019.pdf</a></span></td><td>1.43 MB</td> </tr>
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</div></div></div>Mon, 26 Aug 2019 13:04:14 +0000arash_yavari23532 at https://imechanica.orghttps://imechanica.org/node/23532#commentshttps://imechanica.org/crss/node/23532Compatible-Strain Mixed Finite Element Methods for Incompressible Nonlinear Elasticity
https://imechanica.org/node/22095
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/11923">Mixed finite element methods; finite element exterior calculus; nonlinear elasticity; incompressible elasticity; Hilbert complex</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>We introduce a new family of mixed finite elements for incompressible nonlinear elasticity — <em>compatible-strain mixed finite element methods</em> (CSFEMs). Based on a Hu-Washizu-type functional, we write a four-field mixed formulation with the displacement, the displacement gradient, the first Piola-Kirchhoff stress, and a pressure-like field as the four independent unknowns. Using the Hilbert complexes of nonlinear elasticity, which describe the kinematics and the kinetics of motion, we identify the solution spaces of the independent unknown fields. In particular, we define the displacement in H^1, the displacement gradient in H(curl), the stress in H(div), and the pressure field in L^2. The test spaces of the mixed formulations are chosen to be the same as the corresponding solution spaces. Next, in a conforming setting, we approximate the solution and the test spaces with some piecewise polynomial subspaces of them. Among these approximation spaces are the tensorial analogues of the Nedelec and Raviart-Thomas finite element spaces of vector fields. This approach results in <em>compatible-strain</em> mixed finite element methods that satisfy both the Hadamard compatibility condition and the continuity of traction at the discrete level independently of the refinement level of the mesh. By considering several numerical examples, we demonstrate that CSFEMs have a good performance for bending problems and for bodies with complex geometries. CSFEMs are capable of capturing very large strains and accurately approximating stress and pressure fields. Using CSFEMs, we do not observe any numerical artifacts, e.g., checkerboarding of pressure, hourglass instability, or locking in our numerical examples. Moreover, CSFEMs provide an efficient framework for modeling heterogeneous solids.</p>
</div></div></div><div class="field field-name-upload field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><table class="sticky-enabled">
<thead><tr><th>Attachment</th><th>Size</th> </tr></thead>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://imechanica.org/files/2DMFEMIncmpNLElasFaYa2018.pdf" type="application/pdf; length=2651016">2DMFEMIncmpNLElasFaYa2018.pdf</a></span></td><td>2.53 MB</td> </tr>
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</div></div></div>Tue, 30 Jan 2018 21:58:48 +0000arash_yavari22095 at https://imechanica.orghttps://imechanica.org/node/22095#commentshttps://imechanica.org/crss/node/22095Error | iMechanica