有限元分析结果的可信性直接关系到结构的可靠性，对计算结果进行验证分析已成为当今有限元分析无法回避的问题。近些年，基于flux或flux-free的上下界等验证方法研究已取得了一定的成果，但是非线性输出的验证问题现在仍是有限元分析验证研究的瓶颈，上述方法无法直接使用，因此，对非线性输出验证问题的研究必须另辟蹊径。本项目在申请者前期线性和非线性输出验证研究工作的基础上，将统计物理的系综概念引入到有限元模型中，通过代理方法将原问题转化为由有限元节点构成的非绝缘系综模型，探索有限元模型中的统计行为和特征，建立非绝缘系综变量与有限元模型变量之间的对应关系，构造势能-熵对偶关系替代现行的势能-余能对偶关系，利用熵的极大值可达的性质，研究接触、断裂等问题中精确非线性输出的上下界方法，同时也进一步将该模型和方法扩展到解决其它力学的验证问题。

The trustworthy of Finite element analysis analysis solutions is directly related to the safety of the structures, so it is unavoidable in current finite element analysis that verifications have to be carried out. In recent years, there are some achievements for the flux or flux-free based upper and lower bounding methods for the verification of linear output. Nonlinear output, however, is still a bottle neck to the verification of finite element analysis, the above methods cannot be directly applied to solve it. Therefore, we must find a new way for solving such kind of problems. In this project, depending on the previous research for the verification of linear and nonlinear outputs, we will introduce the concept of ensemble in statistical physics to the finite element model of continuum mechanics, and convert the primal problem to an ensemble model by a surrogate method, establish the relationship between the variables of non-isolated ensemble and finite element model, and by using the dual relation between the potential energy and entropy, explore the statistical behaviour and character in finite element model, and especially with the solvable maximum of entropy, the research on the verifications of nonlinear outputs in fracture and contact problems will be mainly focused on, and further research will be extended to other kind of interface problems.