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Register a new userXFEM based fictitious domain method for linear elasticity model with crack
نهج المجال الخيالي المبني على XFEM لنموذج الالتزام الخطي المتضخم مع الشق
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تقليل تكلفة الحلول الحسابية هو القضية الرئيسية لحل مشكلات التعرف على الشقوق أو الانتشار الشقوق. واحدة من الحلول هي تجنب إعادة الشبكة للنطاق عند تغيير موضع الشق أو عند تمديد الشق. لتجنب إعادة الشبكة، نقترح طريقة العنصر المحدد الخيالي للمحاكاة الرقمية للانقسامات التي تنتجها الشقوق داخل الوسائل المرنة. يستند الأسلوب إلى طريقة النطاق الخيالي الأصلية التي تم تطويرها لشروط ديريشلت لمشكلة بوسون ولمشكلة ستوكس، والتي تم تكيفها مع شروط الحدود النويمان لمشكلات الشقوق. يمثل الشق بواسطة وظائف المستوى. وتم إجراء اختبارات رقمية باستخدام تركيب مختلط للتأكيد على دقة الأسلوب، فضلاً عن ثباته مع الهندسة التي تفرضها تقنية الاستقرار. وتم إثبات شرط التحقق من الأقل من الأعلى لذلك نظرياً. وتم إعطاء تجربة واقعية مع شقة مضغوطة متساوية في الداخل من بركان لإظهار قابلية الأسلوب للتطبيق.
Reduction of computational cost of solutions is a key issue to crack identification or crack propagation problems. One of the solution is to avoid re-meshing the domain when the crack position changes or when the crack extends. To avoid re-meshing, we propose a new finite element approach for the numerical simulation of discontinuities of displacements generated by cracks inside elastic media. The approach is based on a fictitious domain method originally developed for Dirichlet conditions for the Poisson problem and for the Stokes problem, which is adapted to the Neumann boundary conditions of crack problems. The crack is represented by level-set functions. Numerical tests are made with a mixed formulation to emphasize the accuracy of the method, as well as its robustness with respect to the geometry enforced by a stabilization technique. In particular an inf-sup condition is theoretically proven for the latter. A realistic simulation with a uniformly pressurized fracture inside a volcano is given for illustrating the applicability of the method.
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In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by eXtended Finite Element Method and studied for Poisson problem in [Renard]. The method allows computations in domains whose boundaries do not match. A mixed finite element method is used for fluid flow. The interface between the fluid and the structure is localized by a level-set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf-sup condition between the spaces for velocity and the Lagrange multiplier. Convergence analysis is given and several numerical tests are performed to illustrate the capabilities of the method.
In this work we develop a fictitious domain method for the Stokes problem which allows computations in domains whose boundaries do not depend on the mesh. The method is based on the ideas of Xfem and has been first introduced for the Poisson problem. The fluid part is treated by a mixed finite element method, and a Dirichlet condition is imposed by a Lagrange multiplier on an immersed structure localized by a level-set function. A stabilization technique is carried out in order to get the convergence for this multiplier. The latter represents the forces that the fluid applies on the structure. The aim is to perform fluid-structure simulations for which these forces have a central role. We illustrate the capacities of the method by extending it to the incompressible Navier-Stokes equations coupled with a moving rigid solid.
This study suggests a fast computational method for crack propagation, which is based on the extended finite element method (X-FEM). It is well known that the X-FEM might be the most popular numerical method for crack propagation. However, with the increase of complexity of the given problem, the size of FE model and the number of iterative steps are increased correspondingly. To improve the efficiency of X-FEM, an efficient computational method termed decomposed updating reanalysis (DUR) method is suggested. For most of X-FEM simulation procedures, the change of each iterative step is small and it will only lead a local change of stiffness matrix. Therefore, the DUR method is proposed to predict the modified response by only calculating the changed part of equilibrium equations. Compared with other fast computational methods, the distinctive characteristic of the proposed method is to update the modified stiffness matrix with a local updating strategy, which only the changed part of stiffness matrix needs to be updated. To verify the performance of the DUR method, several typical numerical examples have been analyzed and the results demonstrate that this method is a highly efficient method with high accuracy.
In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method, that requires the use of polynomials of degree $kge1$ for stability. Specifically, we show that coercivity can be recovered for $k=0$ by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfillment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy- and the $L^2$-norms of the error, that are shown to convergence, for smooth solutions, as $h$ and $h^2$, respectively (here, $h$ denotes the meshsize). A thorough numerical validation on a complete panel of two- and three-dimensional test cases is provided.
An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The method uses piecewise polynomials of degrees $k (>= 1)$ and $k-1$ respectively for the displacement and stress approximations in the interior of elements inside the subdomains separated by the interface, and piecewise polynomials of degree $k$ for the numerical traces of the displacement on the inter-element boundaries inside the subdomains and on the interface/boundary of the domain. Optimal error estimates in $L^2$-norm for the stress and displacement are derived. Finally, numerical experiments confirm the theoretical results and show that the method also applies to the case of crack-tip domain.
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