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A well-conditioned and optimally convergent XFEM for 3D linear elastic fracture ; ; Bordas, Stéphane et al in International Journal for Numerical Methods in Engineering (n.d.) A variation of the extended finite element method for 3D fracture mechanics is proposed. It utilizes global enrichment and point-wise as well as integral matching of displacements of the standard and ... [more ▼] A variation of the extended finite element method for 3D fracture mechanics is proposed. It utilizes global enrichment and point-wise as well as integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy, optimal convergence rates and improved conditioning for two and three dimensional crack problems. A bespoke benchmark problem is introduced to determine the method's accuracy in the general 3D case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40% for moderately refined meshes and topological enrichment. Moreover, when a fixed enrichment volume is used, the number of iterations required grows at a rate which is reduced by a factor of 2 compared to standard XFEM, diminishing the number of iterations by almost one order of magnitude. [less ▲] Detailed reference viewed: 333 (10 UL)Minimum energy multiple crack propagation. Part II: Discrete Solution with XFEM. Sutula, Danas ; Bordas, Stéphane in Engineering Fracture Mechanics (n.d.) The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼] The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. This Part-II of our three-part paper examines three discrete solution methods for solving fracture mechanics problems based on the principle of minimum total energy. The discrete solution approach is chosen based on the stability property of the fracture configuration at hand. The first method is based on external load-control. It is suitable for stable crack growth and stable fracture configurations. The second method is based on fractured area-control. This method is applicable to stable or unstable fracture growth but it is required that the fracture front be stable. The third solution method is based on a gradient-descent approach. This approach can be applied to arbitrary crack growth problems; however, the gradient-descent formulation cannot be guaranteed to yield the optimal solution in the case of competing crack growth and an unstable fracture front configuration. The main focus is on the gradient-descent solution approach within the framework of the extended finite element discretisation. Although a viable solution method is finally proposed for resolving competing crack growth in the case of an unstable fracture front configuration, the method is not implemented within the present XFEM code but rather exists as a separate proof-of-concept algorithm that is tested against several fabricated benchmark problems. The open-source Matlab code, documentation and example cases are included as supplementary material. [less ▲] Detailed reference viewed: 1657 (129 UL)Minimum energy multiple crack propagation Part I: Theory. Sutula, Danas ; Bordas, Stéphane in Engineering Fracture Mechanics (n.d.) The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼] The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. The key contributions of Part-I of this three-part paper are: (1) formulation of the total energy functional governing multiple crack behaviour, (2) three solution methods to the problem of competing crack growth for different fracture front stabilities (e.g. stable, unstable, or a partially stable configuration of crack tips), and (3) the minimum energy criterion for a set of crack tip extensions is posed as the criterion of vanishing rotational dissipation rates with respect to the rotations of the crack extensions. The formulation lends itself to a straightforward application within a discrete framework for determining the crack extension directions of multiple finite-length crack tip increments, which is tackled in Part-II, using the extended finite element method. In Part-III, we discuss various applications and benchmark problems. The open-source Matlab code, documentation, benchmark/example cases are included as supplementary material. [less ▲] Detailed reference viewed: 3359 (193 UL)Minimum energy multiple crack propagation. Part III: XFEM computer implementation and applications. Sutula, Danas ; Bordas, Stéphane in Engineering Fracture Mechanics (n.d.) The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼] The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. The key contributions of Part-III of the three-part paper are as follows: (1) implementation of XFEM in Matlab with emphasis on the design of the code to enable fast and efficient computational times of fracture problems involving multiple cracks and arbitrary crack intersections, (2) verification of the minimum energy criterion and comparison with the maximum tension criterion via multiple benchmark studies, and (3) we propose a numerical improvement to the crack growth direction criterion that gives significant improvements in accuracy and convergence rates of the fracture paths, especially on coarse meshes. The comparisons of the fracture paths obtained by the maximum tension (or maximum hoop-stress) criterion and the energy minimisation approach via a multitude of numerical case studies show that both criteria converge to virtually the same fracture solutions albeit from opposite directions. In other words, it is found that the converged fracture path lies in between those obtained by each criterion on coarser meshes. Thus, a modified crack growth direction criterion is proposed that assumes the average direction of the directions obtained by the maximum tension and the minimum energy criteria. The numerical results show significant improvements in accuracy (especially on coarse discretisations) and convergence rates of the fracture paths. Finally, the open-source Matlab code, documentation, benchmarks and example cases are included as supplementary material. [less ▲] Detailed reference viewed: 1560 (117 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: 134 (3 UL)A new one point quadrature rule over arbitrary star convex polygon/polyhedron ; ; et al E-print/Working paper (n.d.) The Linear Smoothing (LS) scheme \cite{francisa.ortiz-bernardin2017} ameliorates linear and quadratic approximations over convex polytopes by employing a three-point integration scheme. In this work, we ... [more ▼] The Linear Smoothing (LS) scheme \cite{francisa.ortiz-bernardin2017} ameliorates linear and quadratic approximations over convex polytopes by employing a three-point integration scheme. In this work, we propose a linearly consistent one point integration scheme which possesses the properties of the LS scheme with three integration points but requires one third of the integration computational time. The essence of the proposed technique is to approximate the strain by the smoothed nodal derivatives that are determined by the discrete form of the divergence theorem. This is done by the Taylor's expansion of the weak form which facilitates the evaluation of the smoothed nodal derivatives acting as stabilization terms. The smoothed nodal derivatives are evaluated only at the centroid of each integration cell. These integration cells are the simplex subcells (triangle/tetrahedron in two and three dimensions) obtained by subdividing the polytope. The salient feature of the proposed technique is that it requires only $n$ integrations for an $n-$ sided polytope as opposed to $3n$ in~\cite{francisa.ortiz-bernardin2017} and $13n$ integration points in the conventional approach. The convergence properties, the accuracy, and the efficacy of the LS with one point integration scheme are discussed by solving few benchmark problems in elastostatics. [less ▲] Detailed reference viewed: 68 (3 UL)Coupled Molecular Dynamics and Finite Element Method: simulations of kinetics induced by field mediated interaction ; Baroli, Davide ; et al in Physical Review. E ,Statistical, Nonlinear, and Soft Matter Physics (n.d.) A computational approach coupling Molecular Dynamics (MD)-Finite Element Method (FEM) techniques is here proposed for the theoretical study of the dynamics of particles subjected to the electromechanical ... [more ▼] A computational approach coupling Molecular Dynamics (MD)-Finite Element Method (FEM) techniques is here proposed for the theoretical study of the dynamics of particles subjected to the electromechanical forces. The system consists in spherical particles (modeled as micrometric rigid bodies with proper densities and dielectric functions) suspended in a colloidal solution which flows in a microfluidic channel in the presence of a generic non-uniform variable electric field, generated by electrodes. The particles are subjected to external forces (e.g. drag or gravity) which satisfy the particle-like formulation, typical of the MD approach, and to electromechanical force which in turn needs, during the equation of the motion integration, the self-consistent solutions in three dimensions of correct continuum field equation. In the MD-FEM method used in this work, Finite Element Method is applied to solve the continuum field equation and MD technique is applied to the stepwise explicit integration of equation of the motion. Our work shows the potential of coupled MD-FEM for the study of electromechanical particles and opens the double perspective to use a) MD away from the field of the atomistic simulation and b) the continuum/particle approach to another case where the conventional forces’ evaluation method used in MD is not applicable. [less ▲] Detailed reference viewed: 109 (4 UL)Controlling the Error on Target Motion through Real-time Mesh Adaptation: Applications to Deep Brain Stimulation Bui, Huu Phuoc ; Tomar, Satyendra ; et al E-print/Working paper (n.d.) We present an error-controlled mesh refinement procedure for needle insertion simulation and apply it to the simulation of electrode implantation for deep brain stimulation, including brain shift. Our ... [more ▼] We present an error-controlled mesh refinement procedure for needle insertion simulation and apply it to the simulation of electrode implantation for deep brain stimulation, including brain shift. Our approach enables to control the error in the computation of the displacement and stress fields around the needle tip and needle shaft by suitably refining the mesh, whilst maintaining a coarser mesh in other parts of the domain. We demonstrate through academic and practical examples that our approach increases the accuracy of the displacement and stress fields around the needle without increasing the computational expense. This enables real-time simulations. The proposed methodology has direct implications to increase the accuracy and control the computational expense of the simulation of percutaneous procedures such as biopsy, brachytherapy, regional anesthesia, or cryotherapy and can be essential to the development of robotic guidance. [less ▲] Detailed reference viewed: 475 (34 UL)REMESHED SMOOTHED PARTICLE HYDRODYNAMICS PLATFORM. Obeidat, Anas ; Bordas, Stéphane Poster (n.d.) Detailed reference viewed: 80 (7 UL)Coarsen Graining: A Renewal Concept of Efficient Adaptivity Techniques for Multiscale Models ; ; et al in Computer Methods in Applied Mechanics and Engineering (n.d.) This paper presents a multiscale method for the quasi-static crack propagation. The coarse region is modeled by the di erential reproducing kernel particle(DRKP) method. The coupling between the coarse ... [more ▼] This paper presents a multiscale method for the quasi-static crack propagation. The coarse region is modeled by the di erential reproducing kernel particle(DRKP) method. The coupling between the coarse scale and ne scale is realized through ghost atoms. The ghost atoms positions are interpolated from the coarse scale solution and enforced as boundary conditions on the ne scale. The ne scale region is adaptively enlarged as the crack propagates and the region behind the crack tip is adaptively coarsened. The centro symmetry parameter(CSP) is used to detect the crack tip location. The triangular lattice corresponds to the lattice structure of the (111) plane of an FCC crystal in the ne scale region. The Lennard-Jones potential is used to model the atom-atom interactions. The method is implemented in two dimensions. The results are compared to pure atomistic simulations and show excellent agreement. [less ▲] Detailed reference viewed: 227 (5 UL)Real-time error controlled adaptive mesh refinement in surgical simulation: Application to needle insertion simulation ; Tomar, Satyendra ; et al in IEEE Transactions on Biomedical Engineering (n.d.) This paper presents the first real-time discretisation-error-driven adaptive finite element approach for corotational elasticity problems involving strain localisation. We propose a hexahedron-based ... [more ▼] This paper presents the first real-time discretisation-error-driven adaptive finite element approach for corotational elasticity problems involving strain localisation. We propose a hexahedron-based finite element method combined with local oct-tree $h$-refinement, driven by a posteriori error estimation, for simulating soft tissue deformation. This enables to control the local error and global error level in the mechanical fields during the simulation. The local error level is used to refine the mesh only where it is needed, while maintaining a coarser mesh elsewhere. We investigate the convergence of the algorithm on academic examples, and demonstrate its practical usability on a percutaneous procedure involving needle insertion in a liver. For the latter case, we compare the force displacement curves obtained from the proposed adaptive algorithm with that obtained from a uniform refinement approach. [less ▲] Detailed reference viewed: 658 (63 UL) |
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