有限元计算的面向目标误差估计

Goal-oriented error estimation for linear finite element method

V&V(Verification and Validation),即模型验证与确认,是一种量化复杂数值模拟结果置信度的系统方法.大规模的数值模拟往往不能确保高置信度,数值模拟结果的置信度需要一种严格量化的方法.对于有限元模拟问题,获得近似解后,如果直接对这个解进行误差分析,可以得到一个整体的误差估计.而对于以有限元模拟为辅助手段的设计改进而言,通常都有特别关心的专门设计量,所有的模拟实验过程都是为检验这个量而服务的.面向目标的误差估计方法就是专门针对如何准确和经济地估算特定值误差的一种方法.本文通过线性化简后,把这种估计方法针对有限元模拟成功实现,为在实际工程应用中数值实现这种最接近于解决实际问题的方法作了准备.

VV（verification and validation） are the primary means to access the accuracy and reliability of simulations.A large computer simulation is useful only when reliability of its result is strictly quantitative.For FEM simulation,the result contains all kinds of information.But the designers only care about some special purpose.The aim of the simulation is to check whether a design is fit the purpose.The goal-oriented error estimation approach uses classical duality techniques involving the solution of an adjoint problem to calculate the error of that purpose.In this work,we set up a program for goal-oriented error estimation on linear FEM simulation.It’s a start of practical use for numerical simulation error estimation of engineering applications.