The influence of finite width interfaces on mixed mode stress intensity factors

The problem of a crack approaching a finite width interface is investigated using finite elements. The crack is inclined to the interface and a condition of mixed mode fracture exists. The influence of a finite width bond line interface is considered for one representative material pair combination (E₂/E₁ = 0.10). The stress intensity factors for an inclined crack at various distances from the interface are established as a function of interface width. Maximum cleavage stress and probable angles of crack extension are presented as a function of crack inclination and interface width. Circumferential variations of σ_rr, σ_θθ, and α_rθ are also presented as functions of crack inclination and interface width.     

Detection of cracks in simply-supported beams by continuous wavelet transform of reconstructed modal data

This paper proposes a new approach for damage detection in beam-like structures with small cracks, whose crack ratio [r = H_c/H] is less than 5%, without baseline modal parameters. The approach is based on the difference of the continuous wavelet transforms (CWTs) of two sets of mode shape data which correspond to the left half and the right half of the modal data of a cracked simply-supported beam. The mode shape data of a cracked beam are apparently smooth curves, but actually exhibit local peaks or discontinuities in the region of damage because they include additional response due to the cracks. The modal responses of the damaged simply-supported beams used are computed using the finite element method. The results demonstrate the efficiency of the proposed method for crack detection, and they provide a better crack indicator than the result of the CWT of the original mode shape data. The effects of crack location and sampling interval are examined. The simulated and experimental results show that the proposed method has great potential in crack detection of beam-like structures as it does not require the modal parameter of an uncracked beam as a baseline for crack detection. It can be recommended for real applications.     

Planar crack occupying a finite doubly connected region inside an infinite elastic solid

A method is proposed for the approximate solution of the problem of an embedded pressurized planar crack occupying a finite doubly connected region inside an infinite elastic solid. The formulation of the problem produces a system of two integral equations for determining the unknown normal stresses on the plane of the crack outside the crack region, which can be solved using numerical procedures. The proposed method has been applied to obtain the opening mode stress intensity factors K_I, along the boundary lines of an annular crack subjected to a uniform internal pressure.     

Recursive evaluation of time convolution integrals in the spectral boundary integral method for mode III dynamic fracture problems

The shear crack, propagating spontaneously on a frictional interface, is a useful idealization of a natural earthquake. However, the corresponding boundary value problems are quite challenging in terms of required memory and processor power. While the huge computation amount is reduced by the spectral boundary integral method, the computation effort is still huge. In this paper, a recursive method for the evaluation of convolution integrals was tested in the spectral formulation of the boundary integral method applied to 2D anti-plane crack propagation problems. It is shown that analysis of a 2D anti-plane crack propagation problem involving N_t time steps, based on the recursive evaluation of convolution integrals, requires O(αN_t) computational resources for each Fourier mode (as opposed to O(N_t²) for a classical algorithm), where α is a constant depending on the implementation of the method with typical values much less than N_t. Therefore, this recursive scheme renders feasible investigation of long deformational processes involving large surfaces and long periods of time, while preserving accuracy. The computation methodology implemented here can be extended easily to 3D cases where it can be employed for the simulation of complex spontaneously fault rupture problems which carry a high computational cost.     

The Dirichlet-to-Neumann map for two-dimensional crack problems

A coupled analytic/finite-element method is presented for two-dimensional crack problems. Two classes of problems are studied. The first considers problems where non-linear constitutive processes occur in a region near the crack tip and the remotely applied loading can be characterized by the linear elastic K-field and perhaps the T-stress. In this case, the finite-element method is applied in a circular region around the crack tip where non-linear constitutive response is occurring, and stiffness contributions associated with a numerically implemented Dirichlet-to-Neumann map are imposed on the circular boundary to account for the large surrounding elastic domain and the remote applied loading. The second class of problems considers entirely linear elastic domains with irregular external boundaries and/or complex applied loadings. Here, the discrete Dirichlet-to-Neumann map is used to represent a circular region surrounding the crack tip, and finite-elements are used for the external region. In this case the mixed mode stress intensity factors and the T-stress are retrieved from the map.     

A singular ES-FEM for plastic fracture mechanics

The stress and strain fields around the crack tip for power hardening material, which are singular as r approaches zero, are crucial to fracture and fatigue of structures. To simulate effectively the strain and stress around the crack tip, we develop a seven-node singular element which has a displacement field containing the HRR term and the second order term. The novel singular element is formulated based on the edge-based smoothed finite element method (ES-FEM). With one layer of these singular elements around the crack tip, the ES-FEM works very well for simulating plasticity around the crack tip based on the small strain formulation. Two examples are presented with detailed comparison with other methods. It is found that the results of the presented singular ES-FEM are closer to reference solution, which demonstrates the applicability and the effectiveness of our method for the plastic field around the crack tip.     

A simple approach to the problem of 3-D reconstruction

A relatively simple mathematical procedure for the reconstruction of the 3-dimensional (3D) image of the left ventricle (LV) of the heart is presented. The method is based on the assumption that every ray whoch emanates from the midpoint of the long axis of the 3D body crosses the surface boundary of the ventricle at one and only one point. The coordinates r_i, φ_i, θ_i of the data points on, say, the outer boundary, (i.e., the epicardium) are calculated in a spherical coordinate system having its origin in the midpoint of the long axis. The problem of defining the coordinates of a prescribed grid point on the boundary is treated as an interpolation problem for the function r = r(φ, θ), defined in the rectangle 0 ≤ φ ≤ 2π; 0 ≤ θ ≤ π with r_i given in the points (φ_i, θ_i).     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

An embedded strong discontinuity model for cracking of plain concrete

A numerical model formulated within the framework of a nonsymmetric strong discontinuity approach for fracture simulations of plain concrete is presented. The model is based on the fixed crack concept and makes use of the concept of the elements with embedded discontinuities. Discontinuity segments of individual elements are considered to form a C⁰-continuous path. Enforcement of continuity of the crack path across adjacent elements is established by a partial domain crack tracking algorithm. Orientation of individual crack segments is derived from a nonlocal strain field. The capabilities of the model are shown by means of numerical examples.     

Improvement of the accuracy in boundary element method based on high-order discretization

The boundary element method (BEM) is a popular method to solve various problems in engineering and physics and has been used widely in the last two decades. In high-order discretization the boundary elements are interpolated with some polynomial functions. These polynomials are employed to provide higher degrees of continuity for the geometry of boundary elements, and also they are used as interpolation functions for the variables located on the boundary elements. The main aim of this paper is to improve the accuracy of the high-order discretization in the two-dimensional BEM. In the high-order discretization, both the geometry and the variables of the boundary elements are interpolated with the polynomial function P_m, where m denotes the degree of the polynomial. In the current paper we will prove that if the geometry of the boundary elements is interpolated with the polynomial function P_m+1 instead of P_m, the accuracy of the results increases significantly. The analytical results presented in this work show that employing the new approach, the order of convergence increases from O(L₀)^m to O(L₀)^m+1 without using more CPU time where L₀ is the length of the longest boundary element. The theoretical results are also confirmed by some numerical experiments.     

Bivariate splines for fluid flows

We discuss numerical approximations of the 2D steady-state Navier-Stokes equations in stream function formulation using bivariate splines of arbitrary degree d and arbitrary smoothness r with r<d. We derive the discrete Navier-Stokes equations in terms of B-coefficients of bivariate splines over a triangulation, with curved boundary edges, of any given domain. Smoothness conditions and boundary conditions are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and our numerical experiments show that our method is effective. Numerical simulations of several fluid flows will be included to demonstrate the effectiveness of the bivariate spline method.     

Thermodynamics of horizons in a globally regular spherically symmetric spacetime

We address the question of thermodynamics of horizons in a globally regular spherically symmetric spacetime which is asymptotically de Sitter as r ?? 0 and as r ?? ??. A source term in the Einstein equations smoothly connects two de Sitter vacua with different values of the cosmological constant and corresponds to an anisotropic vacuum fluid defined by symmetry of its stress-energy tensor, which is invariant under radial boosts. In the most general case, the spacetime has three horizons, an internal one, r _a , which is a cosmological horizon for an observer in the R-region 0 ?? r ?? r _a ; the horizon r _b > r _a which is the boundary of the T-region r _a < r < r _b seen as a black or white hole by an observer in the R-region r _b < r < r _c , where r _c is his cosmological horizon. We present a detailed analysis of the thermodynamics of horizons using the Padmanabhan approach relevant to the case of non-zero pressures.     

Continuation Along Bifurcation Branches for a Tumor Model with a Necrotic Core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R and 0<??<R, there exists a radially-symmetric stationary solution with tumor free boundary r=R and necrotic free boundary r=??. The system depends on a positive parameter ??, which describes tumor aggressiveness, and for a sequence of values ?? ₂<?? ₃<??, there exist branches of symmetry-breaking stationary solutions, which bifurcate from these values. Upon discretizing this model, we obtain a family of polynomial systems parameterized by tumor aggressiveness factor???. By continuously changing ?? using a homotopy, we are able to compute nonradial symmetric solutions. We additionally discuss linear and nonlinear stability of such solutions.     

On the equivalence of mode decomposition and mixed finite elements based on the hellinger-reissner principle. Part II: Applications

A general theorem presented in Part I of the paper, specifying necessary and sufficient conditions for the equivalence of the mode decomposition approach and the Hellinger-Reissner mixed method, is used to examine some of the mode decomposition finite elements.The existence of equivalent mixed elements based on the Hellinger-Reissner formulation is established for the class of C⁰ beam elements and a C¹ curved beam element. It is shown that, under the assumptions of the theorem presented in Part I, equivalent Hellinger-Reissner mixed elements do not exist for the existing mode decomposition plate elements.     

An improved kernelization algorithm for r-Set Packing

We present a reduction procedure that takes an arbitrary instance of the r-Set Packing problem and produces an equivalent instance whose number of elements is in O(kr−1), where k is the input parameter. Such parameterized reductions are known as kernelization algorithms, and a reduced instance is called a problem kernel. Our result improves on previously known kernelizations by a factor of k. In particular, the number of elements in a 3-Set Packing kernel is improved from a cubic function of the parameter to a quadratic one.     

Finite elements for elliptic problems with stochastic coefficients

We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen–Loève (KL) expansion. An O(N log N) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M, 1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r = (r₁, …, r_M) in an (M, r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r₁ up to 10 in moderate time.     

Smoothness of Boundaries of Regular Sets

We prove that the boundary of an r-regular set is a codimension one manifold of class C ¹.     

Some numerical investigations in nonlinear optics

The problem of the self-focusing of a light beam in nonlinear media is the central problem in nonlinear optics. The powerful laser beam propagation through a real medium under certain conditions is accompanied with such a phenomenon.Mathematically the problem deals with the investigation of the asymptotic behaviour of the solution of the parabolic equation

$\text{2i}\text{?u}\text{?z}\text{=}\text{?}{}^2\text{u}\text{?r}{}^2\text{+}\text{1}\text{r}\text{?u}\text{?r}\text{+?(∣u∣}{}^2\text{)u}$

with given initial distribution u(r,0) and boundary condition u(∞, z) = 0 where u is the electromagnetic field amplitude, f is a function which describes the refractive index deviation from its constant value in the linear medium. It is complex in the case of nonconservative media. In our investigation we combine analytical and numerical methods. The computational study of the self-focusing problem is complicated due to the boundary condition at infinity and the abrupt light amplitude behaviour in the paraxial region. We managed to overcome these difficulties by introducing the socalled quasi-uniform grid for the radial variable and by using the special technique of the correct transfer of the boundary condition from infinity. The main physical results are: (1) the conditions for the light self-trapping and waveguide creation are found, (2) the self-focusing mechanism and the law of increasing beam amplitude when approaching the collapse point are discovered; (3) the influence of the different kinds of absorption is investigated and the process of light “turbulence” is explained;All the analytical and numerical results are comparable with the experimental situation as well as with treatments by other authors.     

Robust reductions from ranking to classification

We reduce ranking, as measured by the Area Under the Receiver Operating Characteristic Curve (AUC), to binary classification. The core theorem shows that a binary classification regret of r on the induced binary problem implies an AUC regret of at most 2r. This is a large improvement over approaches such as ordering according to regressed scores, which have a regret transform of r ? nr where n is the number of elements.