% CEAD Lab — publications % generated 2026-09-11 @article{jha2026random, title={Continuum Limit of Nonlocal Electrostatics in Random Media}, author={Jha, Prashant K and Dayal, Kaushik}, journal={arXiv preprint arXiv:2609.06354}, year={2026}, url={https://arxiv.org/abs/2609.06354}, doi={10.48550/arXiv.2609.06354} } @article{NANDYALA2026116861, title = {An Information-Theoretic Framework for Optimal Experimental Design in Magnetic Nanoparticle Hyperthermia}, journal = {Applied Mathematical Modelling}, pages = {116861}, year = {2026}, issn = {0307-904X}, doi = {10.1016/j.apm.2026.116861}, url = {https://www.sciencedirect.com/science/article/pii/S0307904X26001228}, author = {Mahesh Nandyala and Andrew Lanham and Prashant K. Jha and Chengyue Wu and John D. Hazle and Thomas E. Yankeelov and Jason Stafford and Ahmed A. El-Gendy and David Fuentes} } @article{galloway2026topology, title={Model-Informed Joint Material-Structural Optimization of Hard-Magnetic Soft Materials}, author={Galloway, Ian and Jha, Prashant K}, journal={arXiv preprint arXiv:2607.14397}, year={2026}, url={https://arxiv.org/abs/2607.14397}, doi={10.48550/arXiv.2607.14397} } @article{jha2026neuraloperators, title={From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing}, author={Jha, Prashant K.}, journal={Mathematics}, volume={14}, number={13}, pages={2421}, year={2026}, publisher={MDPI}, doi={10.3390/math14132421} } @unpublished{enakoutsa2026attention, title={Do attention maps in neural operators learn the Green's function?}, author={Enakoutsa, Koffi and Bogard, Parker and Jha, Prashant K.}, year={2026}, note={Under review, Mathematics and Mechanics of Complex Systems} } @incollection{jha2026nopchapter, title={Neural operators with residual-based correction: Applications to optimization and Bayesian inference}, author={Jha, Prashant K. and Enakoutsa, Koffi}, year={2026}, note={Book chapter, under review} } @article{patra2025predictive, title={Predictive Science: A Quest for the Holy}, author={Patra, Abani and Faghihi, Danial and Jha, Prashant K and Cao, Lianghao and Farrell-Maupin, Kathryn A}, journal={Computing in Science \& Engineering}, year={2026}, volume={28}, number={1}, pages={30--40}, doi = {10.1109/MCSE.2025.3622975}, url = {https://doi.ieeecomputersociety.org/10.1109/MCSE.2025.3622975}, publisher={IEEE} } @article{jha2025peridem, title={PeriDEM -- High-fidelity modeling of granular media consisting of deformable complex-shaped particles}, author={Jha, Prashant K.}, journal={Journal of Open Source Software}, volume={10}, number={110}, pages={7525}, year={2025}, doi={10.21105/joss.07525} } @article{jha2024nodal, title = {Nodal finite element approximation of peridynamics}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {434}, pages = {117519}, year = {2025}, issn = {0045-7825}, doi = {10.1016/j.cma.2024.117519}, url = {https://www.sciencedirect.com/science/article/pii/S0045782524007734}, author = {Prashant K. Jha and Patrick Diehl and Robert Lipton}, keywords = {Nonlocal fracture theory, Peridynamics, Numerical analysis, Finite element method}, abstract = {This work considers the nodal finite element approximation of peridynamics, in which the nodal displacements satisfy the peridynamics equation at each mesh node. For the nonlinear bond-based peridynamics model, it is shown that, under the suitable assumptions on an exact solution, the discretized solution associated with the central-in-time and nodal finite element discretization converges to a solution in the L2 norm at the rate C1Δt+C2h2/ϵ2. Here, Δt, h, and ϵ are time step size, mesh size, and the size of the horizon or nonlocal length scale, respectively. Constants C1 and C2 are independent of h and Δt and depend on norms of the solution and nonlocal length scale. Several numerical examples involving pre-crack, void, and notch are considered, and the efficacy of the proposed nodal finite element discretization is analyzed.} } @article{jha2024residual, title = {Residual-based error corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {419}, pages = {116595}, year = {2024}, issn = {0045-7825}, doi = {10.1016/j.cma.2023.116595}, url = {https://www.sciencedirect.com/science/article/pii/S0045782523007193}, author = {Prashant K. Jha}, keywords = {Neural operators, Operator learning, Singular-value decomposition, Variational formulation, Surrogate modeling, Topology optimization}, abstract = {This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction–diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction–diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.} } @article{jha2022mutual, title={Mutual-information based optimal experimental design for hyperpolarized 13C-pyruvate MRI}, author={Jha, Prashant K and Walker, Christopher and Mitchell, Drew and Oden, J Tinsley and Schellingerhout, Dawid and Bankson, James A and Fuentes, David T}, journal={Scientific reports}, volume={13}, number={1}, pages={18047}, year={2023}, publisher={Nature Publishing Group UK London}, url={https://doi.org/10.1038/s41598-023-44958-y}, doi={10.1038/s41598-023-44958-y} } @article{jha2023continuum, abstract = {We consider electrostatic interactions in two classes of nanostructures embedded in a three dimensional space: (1) helical nanotubes, and (2), thin films with uniform bending (i.e., constant mean curvature). Starting from the atomic scale with a discrete distribution of dipoles, we obtain the continuum limit of the electrostatic energy; the continuum energy depends on the geometric parameters that define the nanostructure, such as the pitch and twist of the helical nanotubes and the curvature of the thin film. We find that the limiting energy is local in nature. This can be rationalized by noticing that the decay of the dipole kernel is sufficiently fast when the lattice sums run over one and two dimensions, and is also consistent with prior work on dimension reduction of continuum micromagnetic bodies to the thin film limit. However, an interesting contrast between the discrete-to-continuum approach and the continuum dimension reduction approaches is that the limit energy in the latter depends only on the normal component of the dipole field, whereas in the discrete-to-continuum approach, both tangential and normal components of the dipole field contribute to the limit energy.}, author = {Jha, Prashant K and Breitzman, Timothy and Dayal, Kaushik}, doi = {10.1007/s00205-023-01869-6}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {2}, pages = {29}, title = {{Discrete-to-Continuum Limits of Long-Range Electrical Interactions in Nanostructures}}, url = {https://doi.org/10.1007/s00205-023-01869-6}, volume = {247}, year = {2023} } @article{cao2022residual, title = {Residual-Based Error Correction for Neural Operator Accelerated Infinite-Dimensional Bayesian Inverse Problems}, journal = {Journal of Computational Physics}, pages = {112104}, year = {2023}, issn = {0021-9991}, doi = {10.1016/j.jcp.2023.112104}, url = {https://www.sciencedirect.com/science/article/pii/S0021999123001997}, author = {Lianghao Cao and Thomas O'Leary-Roseberry and Prashant K. Jha and J. Tinsley Oden and Omar Ghattas}, keywords = {uncertainty quantification, partial differential equations, machine learning, neural networks, operator learning, error analysis}, abstract = {We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.} } @article{jha2022goal, title = {Goal-oriented a-posteriori estimation of model error as an aid to parameter estimation}, journal = {Journal of Computational Physics}, volume = {470}, pages = {111575}, year = {2022}, issn = {0021-9991}, doi = {10.1016/j.jcp.2022.111575}, url = {https://www.sciencedirect.com/science/article/pii/S0021999122006374}, author = {Prashant K. Jha and J. Tinsley Oden}, keywords = {Bayesian inference, A-posterior estimates, Model calibration, Variational formulation, Uncertainty quantification, Goal-oriented a-posterior estimates}, abstract = {In this work, a Bayesian model calibration framework is presented that utilizes goal-oriented a-posterior error estimates in quantities of interest (QoIs) for classes of high-fidelity models characterized by PDEs. It is shown that for a large class of computational models, it is possible to develop a computationally inexpensive procedure for calibrating parameters of high-fidelity models of physical events when the parameters of low-fidelity (surrogate) models are known with acceptable accuracy. The main ingredients in the proposed model calibration scheme are goal-oriented a-posteriori estimates of error in QoIs computed using a so-called lower fidelity model compared to those of an uncalibrated higher fidelity model. The estimates of error in QoIs are used to define likelihood functions in Bayesian inversion analysis. A standard Bayesian approach is employed to compute the posterior distribution of model parameters of high-fidelity models. As applications, parameters in a quasi-linear second-order elliptic boundary-value problem (BVP) are calibrated using a second-order linear elliptic BVP. In a second application, parameters of a tumor growth model involving nonlinear time-dependent PDEs are calibrated using a lower fidelity linear tumor growth model with known parameter values.} } @article{jha2022atomic, author = {Jha, Prashant K. and Marshall, Jason and Knap, Jaroslaw and Dayal, Kaushik}, title = {Atomic-to-Continuum Multiscale Modeling of Defects in Crystals With Nonlocal Electrostatic Interactions}, journal = {Journal of Applied Mechanics}, volume = {90}, number = {2}, year = {2022}, month = {11}, abstract = {This work develops a multiscale modeling framework for defects in crystals with general geometries and boundary conditions in which ionic interactions are important, with potential application to ionic solids and electric field interactions with materials. The overall strategy is posed in the framework of the quasicontinuum multiscale method; specifically, the use of a finite element inspired kinematic description enables a significant reduction in the large number of degrees-of-freedom to describe the atomic positions. The key advance of this work is a method for the efficient and accurate treatment of nonlocal electrostatic charge–charge interactions without restrictions on the geometry or boundary conditions. Electrostatic interactions are long range with slow decay and hence require consideration of all pairs of charges making a brute-force approach computationally prohibitive. The method proposed here accounts for the exact charge–charge interactions in the near-field and uses a coarse-grained approximation in the far-field. The coarse-grained approximation and the associated errors are rigorously derived based on the limit of a finite body with a small periodic lengthscale, thereby enabling the errors in the approximation to be controlled to a desired tolerance. The method is applied to a simple model of gallium nitride, and it is shown that electrostatic interactions can be approximated with a desired level of accuracy using the proposed methodology.}, issn = {0021-8936}, doi = {10.1115/1.4056111}, url = {https://doi.org/10.1115/1.4056111}, note = {021003}, eprint = {https://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/90/2/021003/6945633/jam\_90\_2\_021003.pdf}, } @article{jha2021nlmech, doi = {10.21105/joss.03020}, url = {https://doi.org/10.21105/joss.03020}, year = {2021}, publisher = {The Open Journal}, volume = {6}, number = {65}, pages = {3020}, author = {Prashant K. Jha and Patrick Diehl}, title = {NLMech: Implementation of finite difference/meshfree discretization of nonlocal fracture models}, journal = {Journal of Open Source Software} } @INPROCEEDINGS {gadikar2021load, author = {P. Gadikar and P. Diehl and P. K. Jha}, booktitle = {2021 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)}, title = {Load balancing for distributed nonlocal models within asynchronous many-task systems}, year = {2021}, volume = {}, issn = {}, pages = {669-678}, keywords = {heating systems;distributed processing;computational modeling;partial differential equations;distributed databases;load management;libraries}, doi = {10.1109/IPDPSW52791.2021.00103}, url = {https://doi.ieeecomputersociety.org/10.1109/IPDPSW52791.2021.00103}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, month = {jun} } @article{fritz2021modeling, title = {Modeling and simulation of vascular tumors embedded in evolving capillary networks}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {384}, pages = {113975}, year = {2021}, issn = {0045-7825}, doi = {10.1016/j.cma.2021.113975}, url = {https://www.sciencedirect.com/science/article/pii/S0045782521003066}, author = {Marvin Fritz and Prashant K. Jha and Tobias Köppl and J. Tinsley Oden and Andreas Wagner and Barbara Wohlmuth}, keywords = {Tumor growth, 3D–1D coupled blood flow models, Angiogenesis, Finite elements, Finite volume}, abstract = {In this work, we present a coupled 3D–1D model of solid tumor growth within a dynamically changing vascular network to facilitate realistic simulations of angiogenesis. Additionally, the model includes erosion of the extracellular matrix, interstitial flow, and coupled flow in blood vessels and tissue. We employ continuum mixture theory with stochastic Cahn–Hilliard type phase-field models of tumor growth. The interstitial flow is governed by a mesoscale version of Darcy’s law. The flow in the blood vessels is controlled by Poiseuille flow, and Starling’s law is applied to model the mass transfer in and out of blood vessels. The evolution of the network of blood vessels is orchestrated by the concentration of the tumor angiogenesis factors (TAFs); blood vessels grow towards the increasing TAFs concentrations. This process is not deterministic, allowing random growth of blood vessels and, therefore, due to the coupling of nutrients in tissue and vessels, makes the growth of tumors stochastic. We demonstrate the performance of the model by applying it to a variety of scenarios. Numerical experiments illustrate the flexibility of the model and its ability to generate satellite tumors. Simulations of the effects of angiogenesis on tumor growth are presented as well as sample-independent features of cancer.} } @article{david2021review, author = {Hormuth, David A. and Phillips, Caleb M. and Wu, Chengyue and Lima, Ernesto A. B. F. and Lorenzo, Guillermo and Jha, Prashant K. and Jarrett, Angela M. and Oden, J. Tinsley and Yankeelov, Thomas E.}, title = {Biologically-Based Mathematical Modeling of Tumor Vasculature and Angiogenesis via Time-Resolved Imaging Data}, journal = {Cancers}, volume = {13}, year = {2021}, number = {12}, ARTICLE-NUMBER = {3008}, url = {https://www.mdpi.com/2072-6694/13/12/3008}, issn = {2072-6694}, abstract = {Tumor-associated vasculature is responsible for the delivery of nutrients, removal of waste, and allowing growth beyond 2–3 mm3. Additionally, the vascular network, which is changing in both space and time, fundamentally influences tumor response to both systemic and radiation therapy. Thus, a robust understanding of vascular dynamics is necessary to accurately predict tumor growth, as well as establish optimal treatment protocols to achieve optimal tumor control. Such a goal requires the intimate integration of both theory and experiment. Quantitative and time-resolved imaging methods have emerged as technologies able to visualize and characterize tumor vascular properties before and during therapy at the tissue and cell scale. Parallel to, but separate from those developments, mathematical modeling techniques have been developed to enable in silico investigations into theoretical tumor and vascular dynamics. In particular, recent efforts have sought to integrate both theory and experiment to enable data-driven mathematical modeling. Such mathematical models are calibrated by data obtained from individual tumor-vascular systems to predict future vascular growth, delivery of systemic agents, and response to radiotherapy. In this review, we discuss experimental techniques for visualizing and quantifying vascular dynamics including magnetic resonance imaging, microfluidic devices, and confocal microscopy. We then focus on the integration of these experimental measures with biologically based mathematical models to generate testable predictions.}, doi = {10.3390/cancers13123008} } @article{jha2020peridynamics, title = {Peridynamics-based discrete element method (PeriDEM) model of granular systems involving breakage of arbitrarily shaped particles}, journal = {Journal of the Mechanics and Physics of Solids}, volume = {151}, pages = {104376}, year = {2021}, issn = {0022-5096}, doi = {10.1016/j.jmps.2021.104376}, url = {https://www.sciencedirect.com/science/article/pii/S0022509621000661}, author = {Prashant K. Jha and Prathamesh S. Desai and Debdeep Bhattacharya and Robert Lipton}, keywords = {Peridynamics, Discrete element method, Particle attrition, Particle interlocking, Particle breakage, Granular media, Fracture}, abstract = {Usage, manipulation, transport, delivery, and mixing of granular or particulate media, comprised of spherical or polyhedral particles, is commonly encountered in industrial sectors of construction (cement and rock fragments), pharmaceutics (tablets), and transportation (ballast). Elucidating particulate media’s behavior in concert with particle attrition (i.e., particle wear and subsequent particle fragmentation) is essential for predicting the performance and increasing the efficiency of engineering systems using such media. Discrete element method (DEM) based techniques can describe the interaction between particles but cannot model intra-particle deformation, especially intra-particle fracture. On the other hand, peridynamics provides the means to account for intra-particle deformation and fracture due to contact forces between particles. The present study proposes a hybrid model referred to as PeriDEM that combines the advantages of peridynamics and DEM. The model parameters can be tuned to achieve desired DEM contact forces, damping effects, and intra-particle stiffness. Two particle impacts and compressive behavior of multi-particle systems are thoroughly investigated. The model can account for any arbitrarily shaped particle in general. Spherical, hexagonal, and non-convex particle shapes are simulated in the present study. The effect of mesh resolution on intra-particle peridynamics is explicitly studied. The proposed hybrid model opens a new avenue to explore the complicated interactions encountered in discrete particle dynamics that involve the formation of force chains, particle interlocking, particle attrition, wear, and the eventual breakage.} } @article{fritz2020analysis, title = {Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network}, journal = {Nonlinear Analysis: Real World Applications}, volume = {61}, pages = {103331}, year = {2021}, issn = {1468-1218}, doi = {10.1016/j.nonrwa.2021.103331}, url = {https://www.sciencedirect.com/science/article/pii/S1468121821000432}, author = {Marvin Fritz and Prashant K. Jha and Tobias Köppl and J. Tinsley Oden and Barbara Wohlmuth}, keywords = {Tumor growth, 3D–1D coupled blood flow models, ECM degradation, Existence of weak solutions, Energy inequality, Galerkin method}, abstract = {In this work, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phase-field or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model involves a mesoscale version of Darcy’s law to capture the flow mechanism in the tissue matrix. Modeling flow and transport processes in the vasculature supplying the healthy and cancerous tissue, one-dimensional (1D) equations are considered. Since the models governing the transport and flow processes are defined together with cell species models on a three-dimensional (3D) domain, we obtain a 3D–1D coupled model.} } @article{lipton2020nonlocal, title={Nonlocal elastodynamics and fracture}, author={Lipton, Robert P and Jha, Prashant K}, journal={Nonlinear Differ. Equ. Appl. 28}, year={2021}, volume={23}, doi={10.1007/s00030-021-00683-x} } @article{jha2021finite, title={Finite element approximation of nonlocal dynamic fracture models}, author={Jha, PK and Lipton, R}, journal={Discrete \& Continuous Dynamical Systems-B}, volume={26}, number={3}, pages={1675}, year={2021}, publisher={American Institute of Mathematical Sciences} } @article{diehl2020asynchronous, title={An asynchronous and task-based implementation of peridynamics utilizing HPX—the C++ standard library for parallelism and concurrency}, author={Diehl, Patrick and Jha, Prashant K and Kaiser, Hartmut and Lipton, Robert and L{\'e}vesque, Martin}, journal={SN Applied Sciences}, volume={2}, number={12}, pages={1--21}, year={2020}, publisher={Springer} } @article{Jha2020peri, author = {Jha, Prashant K and Lipton, Robert P}, journal = {International Journal of Fracture}, title = {Kinetic relations and local energy balance for LEFM from a nonlocal peridynamic model}, year = {2020}, doi = {10.1007/s10704-020-00480-0}, url = {https://link.springer.com/article/10.1007/s10704-020-00480-0}, abstract = {A simple nonlocal field theory of peridynamic type is applied to model brittle fracture. The kinetic relation for the crack tip velocity given by Linear Elastic Fracture Mechanics (LEFM) is recovered directly from the nonlocal dynamics, this is seen both theoretically and in simulations. An explicit formula for the change of internal energy inside a neighborhood enclosing the crack tip is found for the nonlocal model and applied to LEFM.} } @article{jha2020bayesian, title={Bayesian-based predictions of COVID-19 evolution in Texas using multispecies mixture-theoretic continuum models}, author={Jha, Prashant K and Cao, Lianghao and Oden, J Tinsley}, journal={Computational Mechanics}, volume={66}, number={5}, pages={1055--1068}, year={2020}, publisher={Springer} } @techreport{oden2020assessment, title={Assessment of Predictability of a Class of Models of Growth of Coronavirus 19 Cases as an Exercise in Predictive Computational Science}, author={Oden, J. Tinsley and Jha, Prashant K. and Cao, Lianghao and Heo, Taemin and Hu, Jing and Hu, Mathew and Kelley, Jonathan and Mora Paz, Jaime D. and Neary, Cyrus and Potla, Akhil and Sherriffdeen, Sheroze and Tessmer, Chase and Yang, Christine}, institution={Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin}, number={20-10}, year={2020}, month={5}, url={https://oden.utexas.edu/media/reports/2020/2010.pdf} } @article{jha2020finite, author = {Jha, Prashant K and Lipton, Robert}, journal = {Communications on Applied Mathematics and Computation}, number = {1}, pages = {93--128}, publisher = {Springer}, title = {Finite element convergence for state-based peridynamic fracture models}, volume = {2}, year = {2020} } @article{lipton2019complex, author = {Lipton, Robert P and Lehoucq, Richard B and Jha, Prashant K}, journal = {Journal of Peridynamics and Nonlocal Modeling}, number = {2}, pages = {122--130}, publisher = {Springer}, title = {Complex fracture nucleation and evolution with nonlocal elastodynamics}, volume = {1}, year = {2019} } @article{jha2019numerical, author = {Jha, Prashant K and Lipton, Robert}, journal = {Computer Methods in Applied Mechanics and Engineering}, pages = {184--225}, publisher = {Elsevier}, title = {Numerical convergence of finite difference approximations for state based peridynamic fracture models}, volume = {351}, year = {2019} } @Inbook{Jha2019wellposedbook, author="Jha, Prashant K. and Lipton, Robert", editor="Voyiadjis, George Z.", title="Well-Posed Nonlinear Nonlocal Fracture Models Associated with Double-Well Potentials", bookTitle="Handbook of Nonlocal Continuum Mechanics for Materials and Structures", year="2019", publisher="Springer International Publishing", address="Cham", pages="1417--1456", abstract="In this chapter, we consider a generic class of bond-based nonlocal nonlinear potentials and formulate the evolution over suitable function spaces. The peridynamic potential considered in this work is a differentiable version of the original bond-based model introduced in Silling (J Mech Phys Solids 48(1):175--209, 2000). The potential associated with the model has two wells where one well corresponds to linear elastic behavior and the other corresponds to brittle fracture (see Lipton (J Elast 117(1):21--50, 2014; 124(2):143--191, 2016)). The parameters in the potential can be directly related to the elastic tensor and fracture toughness. In this chapter we show that well-posed formulations of the model can be developed over different function spaces. Here we will consider formulations posed over H{\"o}lder spaces and Sobolev spaces. The motivation for the H{\"o}lder space formulation is to show a priori convergence for the discrete finite difference method. The motivation for the Sobolev formulation is to show a priori convergence for the finite element method. In the following chapter we will show that the discrete approximations converge to well-posed evolutions. The associated convergence rates are given explicitly in terms of time step and the size of the spatial mesh.", isbn="978-3-319-58729-5", doi="10.1007/978-3-319-58729-5_40", url="https://doi.org/10.1007/978-3-319-58729-5_40" } @Inbook{Jha2019finitebook, author="Jha, Prashant K. and Lipton, Robert", editor="Voyiadjis, George Z.", title="Finite Differences and Finite Elements in Nonlocal Fracture Modeling: A Priori Convergence Rates", bookTitle="Handbook of Nonlocal Continuum Mechanics for Materials and Structures", year="2019", publisher="Springer International Publishing", address="Cham", pages="1457--1494", abstract="In this chapter we present a rigorous convergence analysis of finite difference and finite element approximation of nonlinear nonlocal models. In the previous chapter, we considered a differentiable version of the original bond-based model introduced in Silling (J Mech Phys Solids 48(1):175--209, 2000). There we showed, for a fixed horizon of nonlocal interaction $\epsilon$, that well-posed formulations of the model can be developed over H{\"o}lder spaces and Sobolev spaces. In this chapter we apply these formulations to show a priori convergence for the discrete finite difference and finite element methods. We show that the error made using the forward Euler in time and a finite difference (i.e., piecewise constant) discretization in space with time step $\Delta t$ and spatial discretization h is of the order of $O(\Delta t + h/\epsilon^2)$. For a central difference approximation in time and piecewise linear finite element approximation in space, the approximation error is of the order of $O(\Delta t + h^2/\epsilon^2)$. We point out these are the first such error estimates for nonlinear nonlocal fracture formulations and are reported in Jha and Lipton (2017b Numerical analysis of nonlocal fracture models models in holder space. arXiv preprint arXiv:1701.02818. To appear in SIAM Journal on Numerical Analysis 2018) and Jha and Lipton (2017a, Finite element approximation of nonlocal fracture models. arXiv preprint arXiv:1710.07661). We then go on to prove the stability of the semi-discrete approximation and show that the energy of the discrete approximation is bounded in terms of work done by the body force and initial energy put into the system. We look forward to improvements and development of a posteriori error estimation in the coming years.", isbn="978-3-319-58729-5", doi="10.1007/978-3-319-58729-5_44", url="https://doi.org/10.1007/978-3-319-58729-5_44" } @Inbook{Lipton2018dynamicbook2, author="Lipton, Robert and Said, Eyad and Jha, Prashant K.", editor="Voyiadjis, George Z.", title="Dynamic Damage Propagation with Memory: A State-Based Model", bookTitle="Handbook of Nonlocal Continuum Mechanics for Materials and Structures", year="2019", publisher="Springer International Publishing", address="Cham", pages="1495--1523", abstract="A model for dynamic damage propagation is developed using nonlocal potentials. The model is posed using a state-based peridynamic formulation. The resulting evolution is seen to be well posed. At each instant of the evolution, we identify a damage set. On this set, the local strain has exceeded critical values either for tensile or hydrostatic strain, and damage has occurred. The damage set is nondecreasing with time and is associated with damage state variables defined at each point in the body. We show that a rate form of energy balance holds at each time during the evolution. Away from the damage set, we show that the nonlocal model converges to the linear elastic model in the limit of vanishing nonlocal interaction.", isbn="978-3-319-58729-5", doi="10.1007/978-3-319-58729-5_45", url="https://doi.org/10.1007/978-3-319-58729-5_45" } @Inbook{Lipton2019dynamicbook, author="Lipton, Robert and Said, Eyad and Jha, Prashant K.", editor="Voyiadjis, George Z.", title="Dynamic Brittle Fracture from Nonlocal Double-Well Potentials: A State-Based Model", bookTitle="Handbook of Nonlocal Continuum Mechanics for Materials and Structures", year="2019", publisher="Springer International Publishing", address="Cham", pages="1265--1291", abstract="We introduce a regularized model for free fracture propagation based on nonlocal potentials. We work within the small deformation setting, and the model is developed within a state-based peridynamic formulation. At each instant of the evolution, we identify the softening zone where strains lie above the strength of the material. We show that deformation discontinuities associated with flaws larger than the length scale of nonlocality $\delta$ can become unstable and grow. An explicit inequality is found that shows that the volume of the softening zone goes to zero linearly with the length scale of nonlocal interaction. This scaling is consistent with the notion that a softening zone of width proportional to $\delta$ converges to a sharp fracture set as the length scale of nonlocal interaction goes to zero. Here the softening zone is interpreted as a regularization of the crack network. Inside quiescent regions with no cracks or softening, the nonlocal operator converges to the local elastic operator at a rate proportional to the radius of nonlocal interaction. This model is designed to be calibrated to measured values of critical energy release rate, shear modulus, and bulk modulus of material samples. For this model one is not restricted to Poisson ratios of 1∕4 and can choose the potentials so that small strain behavior is specified by the isotropic elasticity tensor for any material with prescribed shear and Lam{\'e} moduli.", isbn="978-3-319-58729-5", doi="10.1007/978-3-319-58729-5_33", url="https://doi.org/10.1007/978-3-319-58729-5_33" } @article{lipton2018free, author = {Lipton, Robert and Said, Eyad and Jha, Prashant}, journal = {Journal of Elasticity}, number = {2}, pages = {129--153}, publisher = {Springer}, title = {Free damage propagation with memory}, volume = {133}, year = {2018} } @article{jha2018numerical2, author = {Jha, Prashant K and Lipton, Robert}, journal = {International Journal for Numerical Methods in Engineering}, number = {13}, pages = {1389--1410}, publisher = {Wiley Online Library}, title = {Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics}, volume = {114}, year = {2018} } @article{jha2018numerical, author = {Jha, Prashant K and Lipton, Robert}, journal = {SIAM Journal on Numerical Analysis}, number = {2}, pages = {906--941}, publisher = {SIAM}, title = {Numerical analysis of nonlocal fracture models in H{\"o}lder space}, volume = {56}, year = {2018} } @phdthesis{jha2016coarse, title={Coarse Graining of Electric Field Interactions with Materials}, author={Jha, Prashant Kumar}, year={2016}, school={Carnegie Mellon University} }