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We present a new method for the approximate solution of the strongly coupled, nonlinear stress-diffusion problem that appears when modeling hydrogen transport in metals. The most salient feature of the proposed approximation is that it is fully variational, meaning that all the discrete equations are obtained from the optimality conditions of an incremental potential, even for inelastic mechanical behavior. Like other variational methods, the proposed algorithm has remarkable properties, including the symmetry of the tangent operator, making its solution extremely efficient compared to other similar methods available in the literature.

Mathematical and computational tools have proven to be reliable in decision-making processes. In recent times, in particular, machine learning-based methods are becoming increasingly popular as advanced support tools. When dealing with control problems, reinforcement learning has been applied to decision-making in several applications, most notably in games. The success of these methods in finding solutions to complex problems motivates the exploration of new areas where they can be employed to overcome current difficulties. In this paper, we explore the use of automatic control strategies to initial boundary value problems in thermal and disease transport. Specifically, in this work, we adapt an existing reinforcement learning algorithm using a stochastic policy gradient method and we introduce two novel reward functions to drive the flow of the transported field. The new model-based framework exploits the interactions between a reaction-diffusion model and the modified agent. The results show that certain controls can be implemented successfully in these applications, although model simplifications had to be assumed., Comment: 23 pages, 12 figures

We introduce the energy-stepping Monte Carlo (ESMC) method, a Markov chain Monte Carlo (MCMC) algorithm based on the conventional dynamical interpretation of the proposal stage but employing an energy-stepping integrator. The energy-stepping integrator is quasi-explicit, symplectic, energy-conserving, and symmetry-preserving. As a result of the exact energy conservation of energy-stepping integrators, ESMC has a 100\%\ acceptance ratio of the proposal states. Numerical tests provide empirical evidence that ESMC affords a number of additional benefits: the Markov chains it generates have weak autocorrelation, it has the ability to explore distant characteristic sets of the sampled probability distribution and it yields smaller errors than chains sampled with Hamiltonian Monte Carlo (HMC) and similar step sizes. Finally, ESMC benefits from the exact symmetry conservation properties of the energy-stepping integrator when sampling from potentials with built-in symmetries, whether explicitly known or not.

There is increasing recognition that oral health affects overall health and systemic diseases. Nonetheless it remains challenging to rapidly screen patient biopsies for signs of inflammation or the pathogens or foreign materials that elicit the immune response. This is especially true in conditions such as foreign body gingivitis (FBG), where the foreign particles are often difficult to detect. Our long term goal is to establish a method to determine if the inflammation of the gingival tissue is due to the presence of a metal oxide, with emphasis on elements that were previously reported in FBG biopsies, such as silicon dioxide, silica, and titanium dioxide whose persistent presence can be carcinogenic. In this paper, we proposed to use multiple energy X-ray projection imaging to detect and to differentiate different metal oxide particles embedded inside gingival tissues. To simulate the performance of the imaging system, we have used GATE simulation software to mimic the proposed system and to obtain images with different systematic parameters. The simulated parameters include the X-ray tube anode metal, the X-ray spectra bandwidth, the X-ray focal spot size, the X-ray photon number, and the X-ray dector pixel. We have also applied the de-noising algorithm to obtain better Contrast-to-noise ratio (CNR). Our results indicate that it is feasible to detect metal particles as small as 0.5 micrometer in diameter when we use a Chromium anode target with an energy bandwidth of 5 keV, an X-ray photon number of 10^8, and an X-ray detector with a pixel size of 0.5 micrometer and 100 by 100 pixels. We have also found that different metal particles could be differentiated from the CNR at four different X-ray anodes and spectra. These encouraging initial results will guide our future imaging system design., Comment: 28 pages, 12 figures

In the context of mathematical modeling, it is sometimes convenient to integrate models of different nature. These types of combinations, however, might entail difficulties even when individual models are well-understood, particularly in relation to the well-posedness of the ensemble. In this article, we focus on combining two classes of dissimilar diffusive models: the first one defined over a continuum and the second one based on discrete equations that connect average values of the solution over disjoint subdomains. For stationary problems, we show unconditional stability of the linked problems and then the stability and convergence of its discretized counterpart when mixed finite elements are used to approximate the model on the continuum. The theoretical results are highlighted with numerical examples illustrating the effects of linking diffusive models. As a side result, we show that the methods introduced in this article can be used to infer the solution of diffusive problems with incomplete data., Comment: 22 pages, 7 figures, preprint. International Journal for Numerical Methods in Engineering (2023)

In this work, we study lattice structures that exhibit a bistable behavior, i. e., they can snap from one stable state to another, and are also completely reversible, capable of reverting back to its original state through a heat treatment. We design this behavior by constructing lattice structures using networks of nonlinear springs that display tension-compression asymmetry and have different thermal expansion coefficients. The mismatch in the thermal expansion coefficients induces residual stresses in the springs which results in the lattice structure exhibiting bistability at low temperatures and monostability at high temperatures. This behavior mimics the crystallographic phase transformations of shape memory alloys, but here artificially introduced in a structural lattice. By analyzing a representative unit cell, we quantify the effect that the stiffness and the thermal expansion coefficient of the springs have on the stability of the structural lattice. In addition, for simple 2D lattices, using the concept of universal unfoldings of singularity theory, we perform a perturbation analysis to identify the key variables of the structure where controlling defects is important, as they lead to drastic changes in the bifurcation behavior of the lattice. Finally, we verify numerically our analytical predictions in both 2D and 3D simulations using continuation techniques. The examples proposed confirm that the bistable and reversible features of the unit cell carry on to the macroscale, opening the route for the design of lattice structures for energy absorption applications that can hea} with a heat treatment., Comment: 37 pages, 20 figures

The image quality of X-ray imaging relies heavily on the X-ray output number which is dependent on the X-ray tube current and the exposure time. Hybrid X-ray imaging modalities like X-ray luminescence CT (XLCT) and X-ray fluorescence CT (XFCT) rely on the intensity of the X-ray tube to provide an accurate image reconstruction of the nanoprobe distribution in the imaging sample. A limiting factor of good image quality is the radiation dose that will be delivered to the imaging object. To accurately estimate the absorbed dose in an imaging protocol, it is better to simulate the X-ray imaging with a Monte Carlo platform such as GATE (Geant4 Application for Tomographic Emission). However, the input of GATE is a photon number of the simulated X-ray tube. So far, there is no good way to setup the photon number for a desired X-ray tube current. In this work, the accumulated radiation dose of a micro-CT X-ray tube at different current exposure times was recorded with a general-purpose ion chamber. GATE was used to model the total absorbed dose (cGy) in the sensitive volume of the ion chamber with different X-ray output numbers. A linear regression model was generated between the X-ray photon number in the GATE simulations and the tube current exposure time (mAs). The findings of this work provide an approach to correlate the X-ray tube current exposure time (mAs) to the X-ray photon number in GATE simulation of the X-ray tube.

Pencil x-ray beam imaging provides superior spatial resolution than other imaging geometries like sheet beam and cone beam geometries due to the illumination of a line instead of an area or volume. However, the pencil beam geometry suffers from long scan times and concerns over dose discourage laboratory use of pencil beam x-ray sources. Molecular imaging techniques like XLCT imaging benefit most from pencil beam imaging to accurately localize the distribution of contrast agents embedded in a small animal object. To investigate the dose deposited by pencil beam x-ray imaging in XLCT, dose estimations from one angular projection scan by three different x-ray source energies were performed on a small animal object composed of water, bone, and blood with a Monte Carlo simulation platform, GATE (Geant4 Application for Tomographic Emission). Our results indicate that, with an adequate x-ray benchtop source with high brilliance and quasi-monochromatic properties like the Sigray source, the dose concerns can be reduced. With the Sigray source, the bone marrow was estimated to have a radiation dose of 30 mGy for a typical XLCT imaging, in which we have 6 angular projections, 100 micrometer scan step size, and 10^6 x-ray photons per linear scan.

Cone beam computed tomography (CBCT) for breast imaging has potential to replace conventional mammograms. However, concerns over dose and image quality prevent CBBCT systems from the clinical trial phase to next stage. The time of flight (TOF) method was recently shown to reduce the x-ray scattering effects by 95% and improve the image CNR by 110% for large volume objects. The advancements in x-ray sources like in compact Free Electron Lasers (FEL) and advancements in detector technology show potential for the TOF method to be feasible in CBCT when imaging large objects. In this study, we investigate the efficacy of this TOF CBCT in improving the breast cancer imaging. The GATE software was used to simulate the cone beam CT imaging of an 8 cm diameter cylindrical water phantom using a modeled 20 keV quasi-energetic FEL source and various detector temporal resolutions ranging from 1 to 1000 ps. An inhomogeneous breast phantom of similar size was also imaged using the same system setup. Results show that a detector temporal resolution of 10 ps improved the image contrast-to-noise ratio (CNR) by 57% and reduced the scatter-to-primary ratio (SPR) by 8.63 for a small cylindrical phantom. For the breast phantom, the image CNR was enhanced by 12% and the SPR was reduced by 1.35 at 5 ps temporal resolution.

X-ray fluorescence computed tomography (XFCT) is a molecular imaging technique of x-ray photons, which can be used to sense different elements or nanoparticle (NP) agents inside deep samples or tissues. XFCT has been an active research topic for many years. However, XFCT has not been a popular molecular imaging tool because it has limited molecular sensitivity and spatial resolution. To further investigate XFCT imaging, we present a benchtop XFCT imaging system, in in which a unique pencil beam x-ray source and a ring of x-ray spectrometers were simulated using GATE (Geant4 Application for Tomographic Emission) software. An accelerated majorization minimization (MM) algorithm with an L1 regularization scheme was used to reconstruct the XRF image of Molybdenum (Mo) NP targets from the numerical measurements of GATE simulations. With a low x-ray source output rate, good target localization was achieved with a DICE coefficient of 83.681%. The reconstructed signal intensity of the targets was found to be relatively proportional to the target concentrations if detector number and placement is optimized. The MM algorithm performance was compared with maximum likelihood expectation maximization (ML-EM) and filtered back projection (FBP) algorithms. In the future, the benchtop XFCT imaging system will be tested experimentally

We address the formulation and analysis of energy and momentum conserving time integration schemes in the context of particle dynamics, and in particular atomic systems. The article identifies three critical aspects of these models that demand a careful analysis when discretized: first, the treatment of periodic boundary conditions; second, the formulation of approximations of systems with three-body interaction forces; third, their extension to atomic systems with functional potentials. These issues, and in particular their interplay with Energy-Momentum integrators, are studied in detail. Novel expressions for these time integration schemes are proposed and numerical examples are given to illustrate their performance.

This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange-D'Alembert equations are formulated by employing two descriptions, the first one relying on rotations and a splitting approach, and the second one relying on constrained directors. For vanishing external moments, we prove that the new model possesses conservation laws, i.e., the kinetic energy and two nonholonomic momenta that substantially differ from the holonomic momenta preserved by the standard rigid body model. Additionally, we propose a new specialization of a class of energy-momentum integration schemes that exactly preserves the kinetic energy and the nonholonomic momenta replicating the continuous counterpart. Finally, we present numerical results that show the excellent conservation properties as well as the accuracy for the time-discretized governing equations.

We present a new variational principle for linking models of beams and deformable solids, providing also its mathematical analysis. Despite the apparent differences between the two types of governing equations, it will be shown that the equilibrium of systems combining beams and solids can be obtained from a joint constrained variational principle and that the resulting boundary-value problem is well posed.

In all structural models, the section or fiber response is a relation between the strain measures and the stress resultants. This relation can only be expressed in a simple analytical form when the material response is linear elastic. For other, more complex and interesting situations, kinematic and kinetic hypotheses need to be invoked, and a constrained three-dimensional constitutive relation has to be employed at every point of the section in order to implement non-linear and dissipative constitutive laws into dimensionally reduced structural models. In this article we explain in which sense reduced constitutive models can be expressed as minimization problems, helping to formulate the global equilibrium as a single optimization problem. Casting the problem this way has implications from the mathematical and numerical points of view, naturally defining error indicators. General purpose solution algorithms for constrained material response, with and without optimization character, are discussed and provided in an open-source library.

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples., Comment: The paper has been accepted for publication on October 4, 2019

Accurate predictions of thermo-mechanically coupled process in metals can lead to a reduction of cost and an increase of productivity in manufacturing processes such as forming. For modeling these coupled processes with the finite element method, accurate descriptions of both the mechanical and the thermal responses of the material, as well as their interaction, are needed. Conventional material modeling employs empirical macroscopic constitutive relations but does not account for the actual thermo-mechanical mechanisms occurring at the microscopic level. However, the consideration of the latter might be crucial to obtain accurate predictions and a complete understanding of the underlying physics. In this work we describe a fully coupled implicit thermo-mechanical framework for crystal plasticity simulations. This framework includes thermal strains, temperature dependency of the crystal behavior and heat generation by dissipation due to plastic slip and allows the use of large deformation steps thanks to the implicit integration of the governing equations. Its use within computational homogenization simulations allows to bridge the plastic deformation and temperature gradients at the macroscopic scale with the microscopic slip at the grain scale. A series of numerical examples are presented to validate the approach., Comment: submitted to IJP

The present article studies variational principles for the formulation of static and dynamic problems involving Kirchhoff rods in a fully nonlinear setting. These results, some of them new, others scattered in the literature, are presented in a systematic way, helping to clarify certain aspects that have remained obscure. In particular, the study of transversely isotropic models reveals the delicate role that differential geometry plays in their formulation and unveils consequently some approximations that can be made to obtain simplified formulations., Comment: 27 pages, 0 figures

Structure-preserving integrators are in the focus of ongoing research because of their distinguished features of robustness and long time stability. In particular, their formulation for coupled problems that include dissipative mechanisms is still an active topic. Conservative formulations, such as the thermo-elastic case without heat conduction, fit well into a variational framework and have been solved with variational integrators, whereas the inclusions of viscosity and heat transfer are still under investigation. To encompass viscous forces and heat transfer, an extension of Hamilton's principle is required. In this contribution we derive variational integrators for thermo-viscoelastic systems with classical heat transfer. Their results are compared for two discrete model problems vs. Energy-Entropy-Momentum methods. Such comparisons allow to draw conclusions about their relative performance, weaknesses and strengths., Comment: 10 pages, 9 figures, 3 tables