I introduce a new 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 and uniqueness of solutions, upon which we rely to uphold our new method's performance. We present two examples to illustrate the applicability of the method, and to showcase its strengths when compared to the classic Newton-Raphson method, the usual tool of choice in non-linear continuum mechanics.

翻译：本文提出了一种用于解决小应变非线性弹性问题的新型迭代方法。该方法受近期数据驱动计算力学研究的启发，该研究利用"相空间"概念重新定义了连续介质力学中的经典边值问题。相空间是一个抽象度量空间，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都对应其中的一个点。由于相空间与离散化物体相关联，因此它是有限维的。随后定义了两个子集：一个仿射空间称为"物理允许集合"，由满足平衡条件的点构成；另一个"材料允许集合"包含满足本构关系的点。求解边值问题即转化为寻找这两个子域的交集。在线弹性条件下，这可通过求解线性方程组实现；但当材料非线性介入时，情况不再如此，而需要迭代求解方法。我们的迭代方法是在两个集合之间交替投影点，直至收敛。该方法在精神上类似于"交替投影法"和"凸集投影法"，这些方法具有坚实的数学基础，为解的存在性和唯一性提供了条件，我们依托这些条件来支撑新方法的性能。我们通过两个算例展示了该方法的适用性，并突出了其与非线性连续力学中常用工具——经典牛顿-拉夫森方法相比的优势。