Computing upper and lower bounds for the J-integral in two-dimensional linear elasticity
Computing upper and lower bounds for the J-integral in two-dimensional linear elasticity
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DOI：
10.1016/j.cma.2004.12.031
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发表时间：
2006-01
影响因子：
7.2
通讯作者：
Z. Xuan;N. Parés;Jaime Peraire
中科院分区：
文献类型：
--
作者：
Z. Xuan;N. Parés;Jaime Peraire
本文提出了一种计算二维线弹性力学J积分严格上下界的后验方法。J积分通常表示为围道积分，被改写为仅涉及面积积分的位移的二次连续泛函。通过对有限元近似解的二次输出进行展开，将输出表示为已知的可计算量加上解误差的线性和二次泛函。的二次分量是有界的能量范数的误差缩放的连续性常数，这是明确确定的。线性分量表示为位移和计算伴随解中的误差的内积，并且由位移和伴随中的误差的能量范数的适当组合来限定。误差的能量范数的上界是通过使用一个补充能量的方法，需要计算的平衡应力场。用两个平面应变弹性断裂问题说明了该方法。该方法的一个重要特点是，计算的界限是严格的弹性方程的精确弱解。
We present an a posteriori method for computing rigorous upper and lower bounds for the J-integral in two-dimensional linear elasticity. The J-integral, which is typically expressed as a contour integral, is recast as a quadratic continuous functional of the displacement involving only area integration. By expanding the quadratic output about an approximate finite element solution, the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error. The quadratic component is bounded by the energy norm of the error scaled by a continuity constant, which is determined explicitly. The linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution, and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint. Upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields. The method is illustrated with two fracture problems in plane strain elasticity. An important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations.