Kinematic variables bridging discrete and continuum granular mechanics

It is known that there is wide, and at present, unbridgeable, gap between discrete and continuum granular mechanics. In this contribution, first, microscopic kinematic variables neglected in classical continuum granular mechanics are investigated based on the kinematics of discs in contact. Then, a kinematic variable called the averaged pure rotation rate (APR) is proposed for an assembly of circular discs of different sizes, which is then used to produce another two kinematic tensors with one equal to the deformation rate tensor and the other unifying the spin tensor and the APR. As an example, the kinematic variables are incorporated into the unified double-slip plasticity model. Finally, these theoretical analyses are verified using a two-dimensional discrete element method. The study shows that these kinematic variables can be used to bridge discrete and continuum granular mechanics.     

Surface versus bulk nucleation of dislocations during contact

The indentation of single crystals by a periodic array of flat rigid contacts is analyzed using discrete dislocation plasticity. Plane strain analyses are carried out with the dislocations all of edge character and modeled as line singularities in a linear elastic solid. The limiting cases of frictionless and perfectly sticking contacts are considered. The effects of contact size, dislocation source density, and dislocation obstacle density and strength on the evolution of the mean indentation pressure are explored, but the main focus is on contrasting the response of crystals having dislocation sources on the surface with that of crystals having dislocation sources in the bulk. When there are only bulk sources, the mean contact pressure for sufficiently large contacts is independent of the friction condition, whereas for sufficiently small contact sizes, there is a significant dependence on the friction condition. When there are only surface dislocation sources the mean contact pressure increases much more rapidly with indentation depth than when bulk sources are present and the mean contact pressure is very sensitive to the strength of the obstacles to dislocation glide. Also, on unloading a layer of tensile residual stress develops when surface dislocation sources dominate.     

Discrete maximal surfaces with singularities in Minkowski space

We describe discrete maximal surfaces with singularities in 3-dimensional Minkowski space and give a Weierstrass type representation for them. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space generally have certain singularities. We give a criterion that naturally describes the “singular set” for discrete maximal surfaces, including a classification of the various types of singularities that are possible in the discrete case.     

Three solutions to a perturbed nonlinear discrete Dirichlet problem

Using critical point theory, we study the existence of at least three solutions for a perturbed nonlinear Dirichlet boundary value problem for difference equations depending on two positive parameters.     

Haagerup property for C-algebras

We define the Haagerup property for C^?-algebras A and extend this to a notion of relative Haagerup property for the inclusion B⊆A, where B is a C^?-subalgebra of A. Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion A_?α,rΛ⊆A_?α,rΓ has the relative Haagerup property if and only if the quotient group Γ/Λ has the Haagerup property. In particular, the inclusion has the relative Haagerup property if and only if Γ/Λ has the Haagerup property; has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra A(Γ).     

Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.     

Extinction and positivity of the solutions of the heat equations with absorption on networks

In this paper, we propose a discrete version of the following semilinear heat equation with absorption ut=Δu−uq with q>1, which is said to be the ω-heat equation with absorption on a network. Using the discrete Laplacian operator Δω on a weighted graph, we define the ω-heat equations with absorption on networks and give their physical interpretations. The main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish. It is proved that the asymptotic behaviors of the solutions u(x,t) as t tends to +∞ strongly depend on the sign of q−1.     

Diagonalization of tensors with circulant structure

The concepts of tensors with diagonal and circulant structure are defined and a framework is developed for the analysis of such tensors. It is shown a tensor of arbitrary order, which is circulant with respect to two particular modes, can be diagonalized in those modes by discrete Fourier transforms. This property can be used in the efficient solution of linear systems involving contractive products of tensors with circulant structure. Tensors with circulant structure occur in models for image blurring with periodic boundary conditions. It is shown that the new framework can be applied to such problems.     

Efficient pricing of discrete Asian options

Asian options are popular path-dependent financial derivatives. This paper uses lattices to price fixed-strike European-style Asian options that are discretely monitored. The algorithm proposed can also be applied to floating-strike Asian options as well because fixed-strike and floating-strike Asian options are related through an equation. The discretely monitored version is usually found in practice instead of the continuously monitored version usually encountered in the literature. This paper presents the first provably quadratic-time convergent lattice algorithm for pricing fixed-strike European-style discretely monitored Asian options. It is the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to choose the number of states for each node of the lattice. Extensive numerical experiments and comparisons with many existing numerical methods confirm the performance claims and the competitiveness of our algorithm. This result places fixed-strike European-style discretely monitored Asian options in the same complexity class as vanilla options.     

An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates

This paper presents an alternative alpha finite element method using triangular meshes (AαFEM) for static, free vibration and buckling analyses of laminated composite plates. In the AαFEM, an assumed strain field is carefully constructed by combining compatible strains and additional strains with an adjustable parameter α which can produce an effectively softer stiffness formulation compared to the linear triangular element. The stiffness matrices are obtained based on the strain smoothing technique over the smoothing domains and the constant strains on triangular sub-domains associated with the nodes of the elements. The discrete shear gap (DSG) method is incorporated into the AαFEM to eliminate transverse shear locking and an improved triangular element termed as AαDSG3 is proposed. Several numerical examples are then given to demonstrate the effectiveness of the AαDSG3.