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.

Call number: LD2668 .T4 1978 C56 / Master of Science

Call number: LD2668 .T4 1978 L445 / Master of Science

published_or_final_version / abstract / Civil Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Structural reliability technology provides analytical tools for management of uncertainty in all relevant design factors in structural and mechanical systems. Generally, the goal of analysis is to compute probabilities of failure in structural components or system having single or multiple failure mode. Alternately, modern optimization methods provide efficient numerical algorithms for locating optima, particularly in large-scale systems having prescribed deterministic constraints. Optimization procedure can accommodate random variables either directly in its objective function or as one of the primary constraints. The combination of elementary optimization and probabilistic design techniques is the subject of this study. Presented herein is a general strategy for optimization when the design factors are random variables and some or all of the constraints are probability statements. A literature review has indicated that optimization technology in a reliability context has not been fully explored for the general case of nonlinear performance functions and nonnormal variates associated multiple failure modes. This research focuses upon development of the theory to address this general problem. Because analysis algorithms are complicated, a computer code, program RELOPT, is constructed to automate the analysis. The objective function to be minimized is arbitrary, but would generally be the total expected lifetime costs including all initial costs as well as all costs associated with failure. Uncertainty is assumed to be possible in all design factors (including the factors to be determined), and they are modeled as random variables. In general, all of the constraints can be probability statements. The generalized reduce gradient (GRG) method was used for optimization calculations. Options for point probability calculations are first order reliability analysis using the Rackwitz-Fiessler (R-F) or advanced reliability analysis using Wu/FPI. For system reliability analysis either the first order Cornell's bounds or the second order Ditlevsen's bounds can be specified. Several examples are presented to illustrate the full range of capabilities of RELOPT. The program is validated by checking with independent and exact solutions. An example is provided which demonstrates that the cost of running RELOPT can be substantial as the size of the problem increases.

Metallic structures dominated by tensile loads are vulnerable to fatigue and fracture. Fatigue is produced by oscillatory loads. Quasi-static brittle or ductile fracture can result from a "large" load in the random sequence. Moreover, a fatigue or fracture failure in a member of a redundant structure produces impulsive redistributed loads to the intact members. These transient loads could produce a sequence of failures resulting in progressive collapse of the system. Fatigue and fracture design factors are subject to considerable uncertainty. Therefore, a probabilistic approach, which includes a system reliability assessment, is appropriate for design purposes. But system reliability can be improved by a maintenance program of periodic inspection with repair and/or replacement of damaged members. However, a maintenance program can be expensive. The ultimate goal of the engineer is to specify a design, inspection, and repair strategy to minimize life cycle costs. The fatigue/fracture reliability and maintainability (FRM) process for redundant structure can be a complicated random process. The structural model considered series, parallel, and parallel/series systems of elements. Applied to the system are fatigue loads including mean stress, an extreme load, as well as impulsive loads in parallel member systems. The failure modes are fatigue, brittle and ductile fracture. A refined fatigue model is employed which includes both the crack initiation and propagation phases. The FRM process cannot be solved easily using recently developed advanced structural reliability techniques. A "hybrid" simulation method which combines modified importance sampling (MIS) with inflated stress extrapolation (ISE) is proposed. MIS and ISE methods are developed and demonstrated using numerous examples which include series, parallel and series/parallel systems. Not only reasonable estimates of the probability of system failure but also an estimate of the distribution of time to system failure can be obtained. The time to failure distribution can be used to estimate the reliability function, hazard function, conditional reliability given survival at any time, etc. The demonstration cases illustrate how reliability of a system having given material properties is influenced by the number of series and parallel elements, stress level, mean stress, and various inspection/repair policies.

"October 2001." Bibliography: leaves 99-103. Establishes several sets of conditioning relating to the variant optimal deign of linear consecutive-k-out-of-n systems and includes a review of existing research in the theory of variant optimal design of linear consecutive-k-out-of-n systems.

Hydrogen sulfide is toxic by-product of many petroleum, petrochemical and mineral treatment operations. Due to the increasing stringent environment regulations, toxic H2S must be completely removed from industrial waste gases before venting to the atmosphere. The H2S decomposition reaction is a well known thermodynamically limited reaction. Alumina membrane fixed bed catalytic reactors offer the potential for improved conversions at reduced operating temperature due to product separation and catalyst activity. A theoretical and experimental work dealing with a packed bed membrane reactor is the subject of this thesis. A tubular alumina membrane reactor possessing thermal and corrosion resistance has been developed. A multicomponent permeation study indicated that the fluxes of gases could be quantitatively described as a combination of Knudsen diffusion and viscous flow through the porous alumina membrane. The catalytic decomposition of hydrogen sulfide to hydrogen and sulfur was conducted in membrane reactor incorporating a commercial porous alumina membrane in combination with catalytic function of bimetallic RuMo sulfide catalyst. The obtained results demonstrate the possibility of achieving conversion above the equilibrium conversion. The reaction rate is equal to the intrinsic rate since both internal/external mass transfer and heat transfer resistance are negligible for the size of catalyst particles considered. Results obtained with this system have shown a maximum of 2.3 times the equilibrium conversion at the operating temperature 983K, which was equivalent to the conversion at operating temperature 1200K in a conventional fixed bed reactor. The conversion enhancement was significant for the operation with high sweep to feed molar ratio. The reactor configuration of membrane reactor appeared to have an influence on its performance. Comparative experimental and simulation study showed that the cocurrent mode gave slightly higher conversion over counter-current mode. Mathematical models were developed for the reactor, based on plug flow behavior. Simulation had been performed in order to validate the model against experimental data. Reactor optimization was carried out using the validated model. The simulation results from the non-isothermal model were in reasonable agreement with the experimental data. On the other hand, the isothermal model which neglected heat effects that took place in the reactor, has leaded to over-predicted conversion. This study also illustrated that predictive simulations could be used to explore the effects of recycle operation; the optimization study showed that the alumina membrane reactor permitting retentate recycle, could achieve up to 48.6% conversion, corresponding to 6 folded of the equilibrium conversion. The simulations provide a logical methodology for experimental planning and design. To further elucidate the effect of reactor configuration, operation conditions and permeation parameters on the performance of membrane reactors, a high permselective Pt-composite MR model was developed. Comparison of alumina MR and Pt-composite MR was carried out via computer simulation. Porous membrane reactor with higher permeability but lower Permselectivity can attain comparable conversion as the composite membrane reactor with higher permselectivity but lower permeability. Ptcomposite MR was more superior to alumina MR without recycle. Retentate recycle in alumina MR is shown to outperform the Pt-composite MR. Alumina MR was therefore considered as potential candidate for industrial H2S treatment.

Graduation date: 1995