A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics

The finite element (FEM) and the boundary element methods (BEM) are well known powerful numerical techniques for solving a wide range of problems in applied science and engineering. Each method has its own advantages and disadvantages, so that it is desirable to develop a combined finite element/boundary element method approach, which makes use of their advantages and reduces their disadvantages. Several coupling techniques are proposed in the literature, but until now the incompatibility of the basic variables remains a problem to be solved. To overcome this problem, a special super-element using boundary elements based on the usual finite element technique of total potential energy minimization has been developed in this paper. The application of the most commonly used approaches in finite element method namely quarter-point elements and J-integrals techniques were examined using the proposed coupling FEM–BEM. The accuracy and efficiency of the proposed approach have been assessed for the evaluation of stress intensity factors (SIF). It was found that the FEM–BEM coupling technique gives more accurate values of the stress intensity factors with fewer degrees of freedom.     

An adaptive mesh refinement approach for solving non-Hertzian elastic contact problems

Semi-analytical methods are a common way of solving non-hertzian contact problems when designing mechanical components. These methods require of the discretization of the domain into a set of pressure elements and their accuracy and computational cost are related to the number of elements in which the domain is discretized. But, while the accuracy increases as the pressure element mesh is refined, the computational cost increases quadratically with the number of pressure elements. So in the great majority of the cases, a commitment between accuracy and computational cost must be achieved. In this work, a new approach has been developed to improve the performance of semi-analytical methods for solving contact problems. This approach uses an adaptive mesh refinement strategy, based on the quadtree decomposition of the domain. As a result, the computational cost decreases, while the accuracy of the method remains constant.     

求解弹性力学问题的有理单元法

传统的位移有限元法采用多项式形式的位移试函数,对于边数大于4的多边形单元,构造满足单元间协调性要求的多项式形式位移插值函数是一件困难的工作。本文利用逆距离权插值的思想并考虑到单元节点的分布,建立了边数大于4多边形单元上的有理函数形式的形函数。利用有理试函数,采用Galerkin法推导出求解平面弹性力学问题的有理单元法。采用有理单元法求解弹性力学问题,求解区域根据需要可以划分为任意多边形单元,极大地提高了网格划分的灵活性。有理单元法不依赖等参变换,不同单元的形函数表达形式统一,方便计算程序的编写。     

A method for solving periodic contact problems for an elastic strip

Some periodic contact problems for an elastic strip are considered. These problems are solved by the collocation method with the Chebyshev nodes.     

On a class of method for solving problems with random boundary notches and/or cracks

As we know, problems with boundary imperfections (notches or cracks) are the more important one in practical fracture analysis. Frequently, these imperfections would appear in the boundaries of the bodies as a randomly distributed group, and under the loading circumstances would grow up to be unstable cracks which induces catastrophic fracture of the bodies. For right evaluation the fracture behavior of the bodies with such boundary imperfections, it demands mathematical solutions for problems with random boundary notches and/or cracks.     

Non-singular terms for multiple cracks in anisotropic elastic solids

The non-singular and bounded terms for the stresses near the crack tip are investigated. This paper deals with the multiple crack problem for an infinite plate. The original problem is decomposed into three elementary subproblems: (1) the problem for remote uniform stress filed without cracks; (2) the single crack problem with traction applied along the crack face and (3) the problem for the influence of the other cracks. Several examples for collinear and non-collinear cracks are discussed and the results are shown.     

Complementary energy variational approach for plane elastic problems with singularities

Presented is the numerical analysis of plane elastic problems involving stress concentrations and/or singularities using a physically meaningful complementary energy variational approach. The continuum body is modeled by a non-conventional truss structure. Stress distributions in laminated composite bodies and orthotropic sheets with a through crack are obtained. The present results are compared with the analytical solutions for different numerical methods.     

A direct analytic approach for solving two-dimensional linear inverse heat conduction problems

A stable and accurate analytic approach is developed for solving two-dimensional inverse heat conduction problems where the front surface conditions are calculated from the knowledge of the time variation of the temperature at an interior point in the region. A combination of a splitting-up procedure and the least square technique is utilized for the solution of this inverse problem.The stability and accuracy of the present method of analysis are demonstrated by several numerical examples that provided a very strict test. The results illustrate that the method is: insensitive to measurement errors; remains stable for small time steps; and can accomodate, with high degree of accuracy, abrupt change with time in the unknown surface conditions.

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Eine direkte analytische Näherung zur Lösung zweidimensionaler, linear-inverser Wärmeleitungsprobleme

Zusammenfassung Es wurde ein stabiles und genaues analytisches Näherungsverfahren zur Lösung des zweidimensionalen, inversen Wärmeleitungsproblemes entwickelt, wobei die Randbedingungen an der vorderen Oberfläche aus der bekannten, zeitabhängigen Temperaturänderung an einem inneren Punkt in dem Bereich berechnet werden. Es wird eine Kombination einer Aufsplitt-Prozedur und einer Technik des kleinsten Fehlerquadrates benützt, um dieses inverse Problem zu lösen.Die Stabilität und Genauigkeit der vorliegenden Analysenmethode werden durch verschiedene numerische Beispiele demonstriert, die eine sehr strenge Prüfung gestatten. Die Ergebnisse zeigen, daß die Methode unempfindlich gegen Meßfehler ist, für kleine Zeitschritte stabil bleibt und sich mit hohem Auflösungsvermögen an plötzliche, zeitliche Änderungen unbekannter Oberflächenbedingungen anpassen kann.

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Nomenclature a, b lengths of the rectangular region - A ₀,A ₁,A ₂ functions defined by Eq. (6d, e, f) - G ₀,G ₁,₃,P ₀,P ₁,P ₁ p ₂,p ₃,P ₄,P ₅ functions defined in the Appendix - E least square error, Eq. (14) - f(t) actual unknown surface temperature - F (t) f(t)–T /T= modified unknown surface temT perature defined by Eq. (2) - R _t transient solution, Eq. (7) - S _j steady-state solution, Eq. (5) - t time variable, hr. - T temperature variable - T initial and environment temperature - U first-stage solution, Eq. (3) - V second-stage solution, Eq. (10) - x, y rectangular coordinate - y _i simulated experimental data (temperature readings) - Z _i the difference between the applied and computed surface temperature ati ^th time Greek letters thermal diffusivity, m²/h - _j's, _j's coefficients defined in Eq. (2) - _n, _m, _mn eigenvalues defined by Eqs. (6g), (8c, e), respectively - _mn eigenfunctions defined by Eq. (8b) - gq T–T/T, dimensionless temperature - t–_l, time variable, h - ₁ time at which abrupt change takes place in the applied surface temperature, h - standard deviation defined by Eq. (16)     

An approach to solving mechanics problems for materials with multiscale self-similar microstructure

This article proposes an efficient method for solving mechanics boundary value problems formulated for domains with multiscale self-similar microstructure. In particular, composite materials for which one of the phases has a fractal-like structure with scale cut-offs are considered. The boundary value problems are solved using a finite element procedure with enriched shape functions that incorporate information about the geometric complexity. The use of these shape functions makes possible the definition of a unique, parametrically defined model from which the solution for configurations with an arbitrary number of scales can be derived. The proposed method is primarily useful for structures with a large number of self-similar scales for which using the usual finite element method would be too expensive. In order to exemplify the method, a 2D composite with fractal microstructure is considered and several boundary value problems are solved.     

A finite-element approach in stream-tube method for solving fluid and solid mechanics problems

This paper presents a theoretical formulation in which the stream-tube method (STM) is examined through a variational approach for solving solid strain and fluid flow problems with finite elements. The analysis considers a reference domain, used as computational domain, related to the physical domain by an unknown transformation function to be determined numerically. Mass conservation is automatically verified by STM. The variational approach leads to eliminate the pressure in fluid problems and avoids to set up a mixed displacement–pressure procedure in the case of incompressible solids. Examples are given for fluid flows, applications and comparisons are also provided in the bending problem in elasticity.     

A numerical model for the nonlinear interaction of elastic waves with cracks

Microstructures such as cracks and microfractures play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here we simulate wave propagation to understand these mechanisms, complementing existing theoretical and experimental works.We implement two models, one of homogeneous nonlinear elasticity and one of perturbations to cracks, and then use these models to improve our understanding of the relative importance of cracks and intrinsic nonlinearity. We find, by modeling the perturbations in the speed of a low-amplitude P-wave caused by the propagation of a large-amplitude S-wave that the nonlinear interactions of P- and S-waves with cracks are significant when the particle motion is aligned with the normal to the crack face, resulting in a larger magnitude crack dilation. This improves our understanding of the relationship between microstructure orientations and nonlinear wave interactions to allow for better characterization of fractures for analyzing processes including earthquake response, reservoir properties, and non-destructive testing.     

ON A CLASS OF METHOD FOR SOLVING PROBLEMS WITH RANDOM BOUNDARY NOTCHES AND/OR CRACKS(Ⅱ) COMPUTATIONS FOR BOUNDARY NOTCHES

This work is a continuation of the discussion of [1], On a class of method for solving problems with random boundary notches and/or cracks, (I) by C. Ouyang (Appl. Math. & Mech., vol. 1, No. 2, 1980). Here computations for boundary notches are made by using the theory and formulas presented in [1]. In the computation modification is also made for the boundary conditions in parametric plane in [1]. Numerical results for examples show that within ranges of parameter considered in the paper, for example L, the present method in quite workable in practical computations.     

A new approach for solving mixed boundary value problems along holes in orthotropic plates

A new analytical method for solving mixed boundary value problems along holes in composite plates is presented. This addresses problems, where a part of the hole boundary is stress-free, and the other part is subjected to displacement or/and load boundary conditions. The present approach simplifies and speeds up the numerical calculations for the implementation of the boundary conditions by deriving stress functions, which automatically satisfy the stress-free boundary condition over a part of the hole boundary. Only the boundary conditions on the loaded part of the hole need, therefore, to be enforced, typically by a numerical technique such as collocation. Application of this method to orthotropic laminates with a pin-loaded hole problem is discussed.