In this work, we establish a goal-oriented space–time finite element method for a class of dissipative heterogeneous materials. Those materials are modeled on both micro- and macroscale, with a scale transition of volume averaging type satisfying the Hill–Mandel condition. A nonuniform transformation field analysis is performed on the microscopic inelastic strain fields for a model reduction. Reduced variables are deduced from a space–time decomposition of those inelastic strain fields. Closed-form constitutive relations are derived from some dissipative considerations, thus resulting into a reduced order homogenization problem. The resulting model error is sufficiently small for the considered class of materials, thus leaving the discretization error of the finite element method as a main error source. For ease of error estimate, we rewrite the reduced order problem in a multifield formulation. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, rendering error representations of a user-defined quantity of interest. Combining a patch recovery technique, a computable error estimator is developed to quantify both spatial and temporal discretization errors. By means of a localization technique, local error estimators are used to drive a greedy adaptive refinement algorithm in space and time. The effectiveness of the resulting algorithm is confirmed by several numerical examples w.r.t. a prototype model.

在这项工作中，我们为一类耗散异质材料建立了一种面向目标的时空有限元方法。这些材料在微观和宏观尺度上都进行了建模，具有满足 Hill-Mandel 条件的体积平均类型的尺度转变。对微观非弹性应变场进行非均匀变换场分析以简化模型。简化变量是从那些非弹性应变场的时空分解中推导出来的。封闭形式的本构关系源于一些耗散考虑，从而导致降阶同质化问题。对于所考虑的材料类别，由此产生的模型误差足够小，因此留下了有限元法的离散化误差作为主要误差源。为了便于误差估计，我们在多域公式中重写了降阶问题。基于对偶技术，从拉格朗日方法导出时间倒退对偶问题，呈现用户定义的感兴趣量的误差表示。结合补丁恢复技术，开发了一种可计算的误差估计器来量化空间和时间离散化误差。通过定位技术，使用局部误差估计器来驱动空间和时间上的贪婪自适应细化算法。所得算法的有效性通过原型模型的几个数值示例得到证实。