A Total Lagrangian Stochastic Finite Element Method for Hyperelastic Analysis With Uncertainties

Abstract

This article presents a total Lagrangian stochastic finite element method to solve hyperelastic problems with uncertainties. Similar to deterministic hyperelastic analysis, prescribed external forces or boundary values are applied through a series of load steps. By using a stochastic Newton-Raphson method, the stochastic hyperelastic analysis at each load step is linearized as a series of linear stochastic equations. To avoid the use of stochastic configurations from intermediate load steps, all analyses are performed on the initial configuration, thus referred to as a total Lagrangian method. Each stochastic increment of the stochastic solution is then approximated as the product of a random variable and a deterministic vector, and they are solved through a dedicated iteration. Specifically, the deterministic vector is solved using linear deterministic equations, and the corresponding random variable is calculated through one-dimensional stochastic algebraic equations that can be solved efficiently using a sample-based strategy, even for very high-dimensional random inputs. In this way, the proposed method avoids the curse of dimensionality to a great extent. 2D and 3D numerical examples with up to 100 stochastic dimensions demonstrate the promising performance of the proposed method.