Linear smoothed polygonal and polyhedral finite elements

E-print/Working paper (n.d.)

It was observed in [1, 2] that the strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal ... [more ▼]

It was observed in [1, 2] that the strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method. In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes. The main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain. This new strain is then used in the computation of 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 to deliver improved accuracy and pass the patch test to machine precision. [less ▲]

Minimum energy multiple crack propagation. Part III: XFEM computer implementation and applications.

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 ▲]

Linear smoothing over arbitrary polytopes

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 ▲]