On the use of quarter-point tetrahedral finite elements in linear elastic fracture mechanics

This paper discusses the reproduction of the square root singularity in quarter-point tetrahedral (QPT) finite elements. Numerical results confirm that the stress singularity is modeled accurately in a fully unstructured mesh by using QPTs. A displacement correlation (DC) scheme is proposed in combination with QPTs to compute stress intensity factors (SIF) from arbitrary meshes, yielding an average error of 2–3%. This straightforward method is computationally cheap and easy to implement. The results of an extensive parametric study also suggest the existence of an optimum mesh-dependent distance from the crack front at which the DC method computes the most accurate SIFs

A disk-shaped domain integral method for the computation of stress intensity factors using tetrahedral meshes

A novel domain integral approach is introduced for the accurate computation of pointwise J-integral and stress intensity factors (SIFs) of 3D planar cracks using tetrahedral elements. This method is efficient and easy to implement, and does not require a structured mesh around the crack front. The method relies on the construction of virtual disk-shaped integral domains at points along the crack front, and the computation of domain integrals using a series of virtual triangular and line elements. The accuracy of the numerical results computed for through-the-thickness, penny-shaped, and elliptical crack configurations has been validated by using the available analytical formulations. The average error of computed SIFs remains below 1% for fine meshes, and 2–3% for coarse ones. The results of an extensive parametric study suggest that there exists an optimum mesh-dependent domain radius at which the computed SIFs are the most accurate. Furthermore, the results provide evidence that tetrahedral elements are efficient, reliable and robust instruments for accurate linear elastic fracture mechanics calculations

Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra

We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.Comment: 26 pages, 5 figure

Finite element modeling of frictional contact and stress intensity factors in three-dimensional fractured media using unstructured tetrahedral meshes

This thesis introduces a three-dimensional (3D) finite element (FE) formulation to model the linear elastic deformation of fractured media under tensile and compressive loadings. The FE model is based on unstructured meshes using quadratic tetrahedral elements, and includes several novel components: (i) The singular stress field near the crack front is modeled using quarter-point tetrahedral finite elements. (ii) The frictional contact between the crack faces is modeled using isoparametric contact discretization and a gap-based augmented Lagrangian method. (iii) Accurate stress intensity factors (SIFs) of 3D cracks computed using the two novel approaches of displacement correlation and disk-shaped domain integral. The main contributions in the FE modeling of 3D cracks are: (i) It is mathematically proven that quarter-point tetrahedral finite elements (QPTs) reproduce the square root strain singularity of crack problems. (ii) A displacement correlation (DC) scheme is proposed in combination with QPTs to compute SIFs from unstructured meshes. (iii) A novel domain integral approach is introduced for the accurate computation of the pointwise $J$-integral and the SIFs using tetrahedral elements. The main contributions in the contact algorithm are: (i) A square root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. (ii) A gap-based augmented Lagrangian algorithm is introduced for updating the contact forces obtained from the penalty method to more accurate estimates. The results of contact and stress intensity factors are validated for several numerical examples of cubes containing single and multiple cracks. Finally, two applications of this numerical methodology are discussed: (i) Understanding the hysteretic behavior in rock deformation; and (ii) Simulating 3D brittle crack growth. The results in this thesis provide significant evidence that tetrahedral elements are efficient, reliable and robust instruments for accurate linear elastic fracture mechanics calculations.Open Acces

Topological aspect of graphene physics

Topological aspects of graphene are reviewed focusing on the massless Dirac fermions with/without magnetic field. Doubled Dirac cones of graphene are topologically protected by the chiral symmetry. The quantum Hall effect of the graphene is described by the Berry connection of a manybody state by the filled Landau levels which naturally possesses non-Abelian gauge structures. A generic principle of the topologically non trivial states as the bulk-edge correspondence is applied for graphene with/without magnetic field and explain some of the characteristic boundary phenomena of graphene.Comment: 12 pages, 8 figures. Proceedings for HMF-1