Mathematical and numerical studies of shape sensitivity analysis in fracture problems.

应力强度因子（abr）SIF (K)是断裂力学中非常重要的参数，表征为线弹性系统的解（位移）系数。SIF K依赖于材料形状Omega、裂纹形状Sigma和载荷F，即K是{Omega， Sigma， F}的泛函。通过分析计算、数值结果和个案实验对SIF进行了较多的研究，但系统的研究较少。在本研究中，我们推导了SIF对材料形状的形状敏感性分析的表达式。利用作者提出的对偶奇异解技术和gj积分技术对SIF进行表达，得到dR (u， Z) +（边界积分）的r积分表达式。这里u是解，Z是对偶奇异解的正则项，dR是gj积分的r积分（面积积分）的一阶变分。如果解u和解Z在微扰上是正则的，那么我们可以利用gj积分的基本性质将r表达式改为p表达式（线积分），从而阐明形状灵敏度的解析性质。对于数值分析，我们希望使用法国教授Pironneau Olivier等人创建的有限元方法语言的扩展。我们已经添加了函数；面积积分，线积分，光滑截止函数及其偏导数。

Stress intensity factors (abr. SIF) K are very important parameters in fracture mechanics which are characterized as the coefficient of the solution (displacement) of linear elastic system. SIF K depend on the shape of materials Omega, the shape of crack Sigma and the loads F,that is, K is the functional of {Omega, Sigma, F}. There are many researches on SIF by analyitical calculation, numerical results and experiments in individual cases, but systematic researches are few. In this research, we derive the formula which express the shape sensitivity analysis of SIF with respect to the shape of materials Omega. This formula is derived using the expression of SIF by dual singular solution technique and GJ-integral techique proposed by the author, which is given the R-integral expressin dR (u, Z) + (boundary integral). Here u is the solution, Z is the regular term of the dual singular solution, dR is the first variation of R-integral (area integral) of GJ-integral. If the solutions u and Z are regular on the perturbation, then we can change R-expression to P-expression (line-integral) by the fundamental property of GJ-integral that clarify the analytical property of the shape sensitivity. For the numerical analysis, we want to use the extension of the language for finite element method created by Prof. Pironneau Olivier et al. in France. Already we added the functions ; the area integral, line integtal, smooth cut-off functions and its partial derivatives.