Stress intensity factors for elliptical and semi-elliptical cracks subjected to an arbitrary mode l loading

Fatigue durability, damage tolerance and strength evaluations of cracked structural components require accurate determination of stress intensity factors (SIF). Most practical crack configurations are embedded and surface breaking planar cracks subjected to complex two-dimensional stress fields. The only cracked body configuration which has been studied analytically for all types of applied stress fields is a circular crack in an infinite elastic solid. However, this model is suitable only for a narrow class of practical applications. Much wider class of practical problems can be solved using the model of an elliptical crack. The exact analytical SIF solutions for an elliptical crack were obtained only for some particular cases of polynomial applied stress fields. In the present work the exact analytical SIF solution has been obtained for an elliptical crack embedded in an infinite elastic body and subjected to an arbitrary applied normal stress field (Mode I). The most effective method of evaluating the stress intensity factor induced by an applied stress field is by using the weight function for a given cracked body. The weight function represents the SIF induced by a unit concentrated load. The only exact analytical weight function for a planar crack was obtained for a circular one. In the present research the exact analytical weight function has been derived for an elliptical crack embedded in an infinite elastic solid. The weight function for an elliptical crack was subsequently employed in the alternating method to obtain the unique SIF solution for a surface breaking semi-elliptical crack in a semi-infinite body subjected to an arbitrary applied stress field. The solutions obtained in the present work can be used in various practical applications, such as cracks in pressure vessels, welded structures and mechanical engineering components subjected to cyclic loading

Adaptive shape optimization with NURBS designs and PHT-splines for solution approximation in time-harmonic acoustics

Geometry Independent Field approximaTion (GIFT) was proposed as a generalization of Isogeometric analysis (IGA), where different types of splines are used for the parameterization of the computational domain and approximation of the unknown solution. GIFT with Non-Uniform Rational B-Splines (NUBRS) for the geometry and PHT-splines for the solution approximation were successfully applied to problems of time-harmonic acoustics, where it was shown that in some cases, adaptive PHT-spline mesh yields highly accurate solutions at lower computational cost than methods with uniform refinement. Therefore, it is of interest to investigate performance of GIFT for shape optimization problems, where NURBS are used to model the boundary with their control points being the design variables and PHT-splines are used to approximate the solution adaptively to the boundary changes during the optimization process. In this work we demonstrate the application of GIFT for 2D acoustic shape optimization problems and, using three benchmark examples, we show that the method yields accurate solutions with significant computational savings in terms of the number of degrees of freedom and computational time

A one point integration rule over star convex polytopes

In this paper, the recently proposed linearly consistent one point integration rule for the meshfree methods is extended to arbitrary polytopes. The salient feature of the proposed technique is that it requires only one integration point within each n-sided polytope as opposed to 3n in Francis et al. (2017) and 13n integration points in the conventional approach for numerically integrating the weak form in two dimensions. The essence of the proposed technique is to approximate the compatible strain by a linear smoothing function and evaluate the smoothed nodal derivatives by the discrete form of the divergence theorem at the geometric center. This is done by Taylor's expansion of the weak form which facilitates the use of the smoothed nodal derivatives acting as the stabilization term. This translates to 50% and 30% reduction in the overall computational time in the two and three dimensions, respectively, whilst preserving the accuracy and the convergence rates. Th

Application of PHT-splines in bending and vibration analysis of cracked Kirchhoff–Love plates

In this work, we present an eXtended Geometry Independent Field approximaTion (X–GIFT) formulation for cracked Kirchhoff–Love plates. The plate geometry is modeled by Non-Uniform Rational B-Splines (NURBS) while the solution is approximated by Polynomial Splines over Hierarchical T-meshes (PHT-splines) and enriched by the Heaviside function and crack tip asymptotic expansions. The adaptive refinement is driven by a recovery-based error estimator. The formulation is employed for bending and vibration analysis. We compare different strategies for refinement, enrichment and evaluation of fracture parameters. The obtained results are shown to be in a good agreement with the reference solutions

A new one point quadrature rule over arbitrary star convex polygon/polyhedron

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.

Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)

This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution field. The method inherits the main advantage of isogeometric analysis, i.e. preserves the original, exact CAD geometry (for example, given by NURBS), but allows pairing it with an approximation space which is more suitable/flexible for analysis, for example, T-splines, LR-splines, (truncated) hierarchical B-splines, and PHT-splines. This generalization offers the advantage of adaptive local refinement without the need to re-parameterize the domain, and therefore without weakening the link with the CAD model. We demonstrate the use of the method with different choices of the geometry and field splines, and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for non-nested spaces