Computer simulations for civil and mechanical engineering that efficiently leverage computational resources to solve boundary value problems have pervasive impacts on many aspects of civilization, including manufacturing, communication, transportation, medicine, and defense. Conventionally, a solver that predicts the mechanical behaviors of solids requires constitutive laws that represent mechanisms not directly derived from balance principles. These mechanisms are often characterized by mathematical models validated and tested via tabulated data, organized in grids or, more broadly, within normed Euclidean space (e.g., principle stress space, Mohr circle). These mathematical models often involved mapping between/among normed vector spaces that adhere to physical constraints. This methodology has manifested frameworks such as hyperelastic energy functional, elastoplasticity models with evolving internal variables, cohesive zone models for fracture, etc. However, the geometry of material data plays a crucial role in the efficiency, accuracy, and robustness of predictions.
This thesis introduces a collection of mathematical models, tools, algorithms, and frameworks that, when integrated, may unleash the potential of leveraging data geometry to advance solid mechanics modeling. In the first part of the thesis, we introduce the concept of treating constitutive data as a manifold. This idea leads to a novel data-driven paradigm called “Manifold Embedding Data-Driven Mechanics,” which incorporates the manifold structure of data into the distance minimization model-free method. By training an invertible artificial neural network (ANN) to embed nonlinear constitutive data onto a hyperplane, we replace the costly combinatoric optimization necessary for the classical model-free paradigm with a projection and, as a result, significantly improve the efficiency and robustness of the model-free approach with a distance measure consistent with the data geometry. This method facilitates consistent interpolation on the manifold, which improves the accuracy when data is limited. To handle noisy data, we relax the invertibility constraint of the designed ANN and construct the desired embedding space via a geometric autoencoder. Unlike the classical autoencoder, which compresses data by reducing the data dimensionality in the latent space, our design focuses on reducing the dimensionality of the data by imposing constraints. This technique enables us to learn a noise-free embedding through a simple projection by assuming the orthogonality between the data and noise.
To improve the interpretability and, ultimately, the trustworthiness of machine learning-derived constitutive models, we abandon the design of the fully connected neural networks and instead introduce polynomials in feature space that enable us to turn neural network parametrized black-box models back into mathematical models understandable by engineers. We present geometrically inspired structures in a feature space spanned by univariate ANNs and then learn a sparse representation of the data using these acquired features. Our divide-and-conquer scheme takes advantage of the learned univariate functions to perform parallel symbolic regression, ultimately extracting human-readable equations for material modeling. Our approach mitigates the well-known computational burden associated with symbolic regression for high-dimensional data and data that must adhere to physical constraints. We demonstrate the interpretability, accuracy, and computational efficiency of our algorithm in discovering constitutive models for hyperelastic materials and plastic yield surfaces.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/x6qt-4981 |
| Date | January 2024 |
| Creators | Bahmani, Bahador |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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