Abstract
Accurate prediction of the mechanical failure of structural components plays an important role in the design of engineering structures. However, the fracture process is challenging to model numerically due to the existence of an elastic singularity at the crack tip. The phase field model (PFM) is one promising approach for modeling brittle fracture using an auxiliary field variable to regularize discontinuities associated with sharp cracks. Unfortunately, it is generally computationally expensive due to the need to solve a fourth-order partial differential equation. In order to reduce the computational burden for parameterized problems, different reduced order modeling approaches such as proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM), have been proposed. However, these frameworks are model-based and intrusive approaches and need in-depth efforts in modifying existing complex simulation codes. In this work, a tensor train (TT) approach is proposed in combination with Gaussian process regression (GPR) for modeling and predicting the dynamics of fracture in composite materials. As opposed to POD and DEIM, the TT-GPR approach is a fully data-driven and non-intrusive approach. In this work, we study the fracture of a brittle elastic 2D rectangular slab with a pre-existing crack under Mode I (crack opening) loading conditions. The high dimensional training data for the TT model was generated using PFM with the finite difference method. The predictions by the TT-GPR model are compared with the results from the finite difference method. The TT-GPR is robust enough to predict the catastrophic evolution of the crack growth trend with an accuracy of up to 95% with computational load reduced by two orders of magnitude.
Highlights
While the prediction of mechanical failure due to fracture plays a crucial role in structural and mechanical engineering design, the search for a numerical model that can capture the details of crack propagation can be quite challenging
In order to reduce the computational burden for parameterized problems, different reduced order modeling approaches such as proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM), have been proposed
A tensor train (TT) approach is proposed in combination with Gaussian process regression (GPR) for modeling and predicting the dynamics of fracture in composite materials
Summary
While the prediction of mechanical failure due to fracture plays a crucial role in structural and mechanical engineering design, the search for a numerical model that can capture the details of crack propagation can be quite challenging. Intrusive model-based approaches such as proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) are normally used for model reduction of a parameterized problem [5, 6]. Due to their intrusive nature, they required modification of the complex simulation codes that describe the model. This may in turn reduce the numerical stability and capability of the prediction model due to the vectorization of multidimensional solutions that are required for deriving the POD/ DEIM modes [7].
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