Modeling of cortical bone adaptation due to oscillatory loading

Bone is a living tissue which constantly undergoes a complex process of adaptation in response to its biochemical and mechanical environment to optimize its resistance to failure. The bone adaptation due to the mechanical loading is dependent on a combination of different mechanical stimuli such as the magnitude and frequency of the applied load, number of cycles, number of bouts, time between bounds, and other factors. In this presentation we discuss the model of adaptation in cortical bone which employs the finite element stress analysis coupled with an evolution law. The finite element model is generated from microcomputed tomography images of the rat ulna and the stress analysis is carried out using boundary and loading conditions on the rat ulna obtained from the experiments of Robling et al. [1]. Initially, we use an elastic material model and a simple growth law with strain energy density as the mechanical stimulus to simulate the effects of the load magnitude and the number of bouts. Then, we include the effect of load induced fluid flow in cortical bone by modeling bone as a poroelastic material. Our analysis focuses on the growth behavior in a rat ulna due to oscillatory loading. We use the dissipation energy based the mechanical stimulus as the triggering stimulus for adaptation using arguments based on the second law of thermodynamics. In the analysis, we account for the hierarchical structure of bone indirectly. Very good agreement is found with the experiments of Robling et al. [1]. REFERENCE [1] Robling et al. JBMR, 2002, 17, 1545

Dedication

n/

On the use of graph neural networks and shape-function-based gradient computation in the deep energy method

A graph neural network (GCN) is employed in the deep energy method (DEM) model to solve the momentum balance equation in 3D for the deformation of linear elastic and hyperelastic materials due to its ability to handle irregular domains over the traditional DEM method based on a multilayer perceptron (MLP) network. Its accuracy and solution time are compared to the DEM model based on a MLP network. We demonstrate that the GCN-based model delivers similar accuracy while having a shorter run time through numerical examples. Two different spatial gradient computation techniques, one based on automatic differentiation (AD) and the other based on shape function (SF) gradients, are also accessed. We provide a simple example to demonstrate the strain localization instability associated with the AD-based gradient computation and show that the instability exists in more general cases by four numerical examples. The SF-based gradient computation is shown to be more robust and delivers an accurate solution even at severe deformations. Therefore, the combination of the GCN-based DEM model and SF-based gradient computation is potentially a promising candidate for solving problems involving severe material and geometric nonlinearities

Effect of Type 1 Diabetes and Age on Hydroxyapatite Concentration and Elastic Modulus in the Cortical Bone

It is common for patients with type 1 diabetes to have marked reductions in body strength and ability to perform physical activities. These markers for physical strength are also known to change with age. Type 1 diabetes is correlated with fatigue but is not known to be related to changes in the cortical bone. In this study, we hypothesize that type 1 diabetes causes a reduction in the concentration of hydroxyapatite (HA) and in the elastic modulus of the cortical bone. Moreover, we predict that adult-aged bones will exhibit greater hydroxyapatite concentrations and elastic moduli than younger bones. We perform this study on Lewis rats with and without type 1 diabetes and at different ages. The tibiae and femora of each rat have been separated from the body and isolated for medical imaging. We determine the hydroxyapatite content of the bones by analyzing images collected with a microCT scanner and comparing the radiodensities of these bones to phantoms of known hydroxyapatite content. Elastic moduli data are also collected by performing a mechanical bending test on each bone. Through our experimental data, we hope to determine how type 1 diabetes and age, when considered separately, affect the parameters of interest.Ope

Effect of Type 1 Diabetes and Age on Hydroxyapatite Concentration and Elastic Modulus in the Cortical Bone

It is common for patients with type 1 diabetes to have marked reductions in body strength and ability to perform physical activities. These markers for physical strength are also known to change with age. Type 1 diabetes is correlated with fatigue but is not known to be related to changes in the cortical bone. In this study, we hypothesize that type 1 diabetes causes a reduction in the concentration of hydroxyapatite (HA) and in the elastic modulus of the cortical bone. Moreover, we predict that adult-aged bones will exhibit greater hydroxyapatite concentrations and elastic moduli than younger bones. We perform this study on Lewis rats with and without type 1 diabetes and at different ages. The tibiae and femora of each rat have been separated from the body and isolated for medical imaging. We determine the hydroxyapatite content of the bones by analyzing images collected with a microCT scanner and comparing the radiodensities of these bones to phantoms of known hydroxyapatite content. Elastic moduli data are also collected by performing a mechanical bending test on each bone. Through our experimental data, we hope to determine how type 1 diabetes and age, when considered separately, affect the parameters of interest.Ope

Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads

A novel deep operator network (DeepONet) with a residual U-Net (ResUNet) as the trunk network is devised to predict full-field highly nonlinear elastic-plastic stress response for complex geometries obtained from topology optimization under variable loads. The proposed DeepONet uses a ResUNet in the trunk to encode complex input geometries, and a fully-connected branch network encodes the parametric loads. Additional information fusion is introduced via an element-wise multiplication of the encoded latent space to improve prediction accuracy further. The performance of the proposed DeepONet was compared to two baseline models, a standalone ResUNet and a DeepONet with fully connected networks as the branch and trunk. The results show that ResUNet and the proposed DeepONet share comparable accuracy; both can predict the stress field and accurately identify stress concentration points. However, the novel DeepONet is more memory efficient and allows greater flexibility with framework architecture modifications. The DeepONet with fully connected networks suffers from high prediction error due to its inability to effectively encode the complex, varying geometry. Once trained, all three networks can predict the full stress distribution orders of magnitude faster than finite element simulations. The proposed network can quickly guide preliminary optimization, designs, sensitivity analysis, uncertainty quantification, and many other nonlinear analyses that require extensive forward evaluations with variable geometries, loads, and other parameters. This work marks the first time a ResUNet is used as the trunk network in the DeepONet architecture and the first time that DeepONet solves problems with complex, varying input geometries under parametric loads and elasto-plastic material behavior

A deep learning energy-based method for classical elastoplasticity

The deep energy method (DEM) has been used to solve the elastic deformation of structures with linear elasticity, hyperelasticity, and strain-gradient elasticity material models based on the principle of minimum potential energy. In this work, we extend DEM to elastoplasticity problems involving path dependence and irreversibility. A loss function inspired by the discrete variational formulation of plasticity is proposed. The radial return algorithm is coupled with DEM to update the plastic internal state variables without violating the Kuhn-Tucker consistency conditions. Finite element shape functions and their gradients are used to approximate the spatial gradients of the DEM-predicted displacements, and Gauss quadrature is used to integrate the loss function. Four numerical examples are presented to demonstrate the use of the framework, such as generating stress-strain curves in cyclic loading, material heterogeneity, performance comparison with other physics-informed methods, and simulation/inference on unstructured meshes. In all cases, the DEM solution shows decent accuracy compared to the reference solution obtained from the finite element method. The current DEM model marks the first time that energy-based physics-informed neural networks are extended to plasticity, and offers promising potential to effectively solve elastoplasticity problems from scratch using deep neural networks

Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads

Deep Operator Network (DeepONet), a recently introduced deep learning operator network, approximates linear and nonlinear solution operators by taking parametric functions (infinite-dimensional objects) as inputs and mapping them to solution functions in contrast to classical neural networks that need re-training for every new set of parametric inputs. In this work, we have extended the classical formulation of DeepONets by introducing sequential learning models like the gated recurrent unit (GRU) and long short-term memory (LSTM) in the branch network to allow for accurate predictions of the solution contour plots under parametric and time-dependent loading histories. Two example problems, one on transient heat transfer and the other on path-dependent plastic loading, were shown to demonstrate the capabilities of the new architectures compared to the benchmark DeepONet model with a feed-forward neural network (FNN) in the branch. Despite being more computationally expensive, the GRU- and LSTM-DeepONets lowered the prediction error by half (0.06\% vs. 0.12\%) compared to FNN-DeepONet in the heat transfer problem, and by 2.5 times (0.85\% vs. 3\%) in the plasticity problem. In all cases, the proposed DeepONets achieved a prediction $R^2$ value of above 0.995, indicating superior accuracy. Results show that once trained, the proposed DeepONets can accurately predict the final full-field solution over the entire domain and are at least two orders of magnitude faster than direct finite element simulations, rendering it an accurate and robust surrogate model for rapid preliminary evaluations