We introduce an iterative method to solve problems in small-strain non-linear elasticity. The method is inspired by recent work in data-driven computational mechanics, which reformulated the classic boundary value problem of continuum mechanics using the concept of "phase space". The latter is an abstract metric space, whose coordinates are indexed by strains and stress components, where each possible state of the discretized body corresponds to a point. Since the phase space is associated to the discretized body, it is finite dimensional. Two subsets are then defined: an affine space termed "physically-admissible set" made up by those points that satisfy equilibrium and a "materially-admissible set" containing points that satisfy the constitutive law. Solving the boundary-value problem amounts to finding the intersection between these two subdomains. In the linear-elastic setting, this can be achieved through the solution of a set of linear equations; when material non-linearity enters the picture, such is not the case anymore and iterative solution approaches are necessary. Our iterative method consists on projecting points alternatively from one set to the other, until convergence. The method is similar in spirit to the ``method of alternative projections'' and to the ``method of projections onto convex sets'', for which there is a solid mathematical foundation that furnishes conditions for existence of solutions and convergence, upon which we rely to assess the method's performance. We present a proof of geometric convergence rate and two examples: a fundamental one to illustrate the features of the method, and a realistic one to showcase its capacities and strengths when compared to the classic Newton-Raphson method, the usual tool of choice in non-linear continuum mechanics. Finally, its ability to deal with constitutive laws based on neural network is also showcased.

翻译：本文提出了一种求解小应变非线性弹性问题的迭代方法。该方法受数据驱动计算力学近期研究的启发，该研究利用"相空间"概念重构了连续介质力学的经典边值问题。相空间是一个抽象的度量空间，其坐标由应变和应力分量索引，离散化物体的每个可能状态对应空间中的一个点。由于相空间与离散化物体相关联，因此是有限维的。随后定义两个子集：由满足平衡条件的点构成的仿射空间称为"物理容许集"，以及包含满足本构关系点的"材料容许集"。求解边值问题等价于寻找这两个子域的交集。在线弹性情况下，这可通过求解线性方程组实现；当涉及材料非线性时，情况发生变化，必须采用迭代求解方法。本文迭代方法的核心在于交替地将点从一个集合投影至另一个集合，直至收敛。该方法在本质上类似于"交替投影法"和"凸集投影法"，这两种方法具有坚实的数学基础，能够提供解的存在性和收敛性条件，我们依此评估该方法的性能。我们给出了几何收敛速度的证明和两个算例：基础算例用于说明方法的特性，实际算例则展示其相较于非线性连续介质力学常用工具——经典牛顿-拉弗森方法的性能优势。最后，还展示了该方法处理基于神经网络的本构关系的能力。