@article{hu_smith_burch_hays_oates_2014, title={A modeling and uncertainty quantification framework for a flexible structure with macrofiber composite actuators operating in hysteretic regimes}, volume={25}, ISSN={["1530-8138"]}, DOI={10.1177/1045389x13489781}, abstractNote={Macrofiber composites are low cost, durable, and flexible piezoceramic devices that are presently being considered for applications that include shape control of airfoils for improved flight performance, vibration, and noise suppression and energy harvesting. However, macrofiber composites also exhibit hysteresis and constitutive nonlinearities that need to be incorporated in models and model-based control designs to achieve their full capability. In this article, we combine constitutive relations, constructed using the homogenized energy model for ferroelectric hysteresis, with Euler–Bernoulli theory to construct a dynamic macrofiber composite model that quantifies a range of rate-dependent hysteretic behavior of macrofiber composites. Using homogenizing strategies, the macrofiber composite patch is treated as a monolithic material with effective parameters. We initially calibrate the model by estimating parameters through a least squares fit to a subset of the measured data. We find that the estimated parameters yield very accurate fits for quasi-static hysteresis. The estimated parameters also provide reasonably accurate predictions for a range of frequencies that include the first two harmonics. Second, we employ an adaptive Markov chain Monte Carlo algorithm to construct densities and analyze the correlation between parameters. The kernel density estimates derived from the Markov chain Monte Carlo chains imply that most of the model parameters exhibit non-Gaussian distributions.}, number={2}, journal={JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES}, author={Hu, Zhengzheng and Smith, Ralph C. and Burch, Nathanial and Hays, Michael and Oates, William S.}, year={2014}, month={Jan}, pages={204–228} }
 @article{hu_smith_willert_kelley_2014, title={High-Dimensional Model Representations for the Neutron Transport Equation}, volume={177}, ISSN={["1943-748X"]}, DOI={10.13182/nse13-52}, abstractNote={Abstract The Boltzmann transport equation is used to model the neutron flux in a nuclear reactor. The solution of the transport equation is the neutron flux, which depends on a large number of material cross sections that can be on the order of thousands. These cross sections describe various types of possible interactions between neutrons, such as fission, capture, and scattering. The cross sections are measured experimentally and therefore have associated uncertainties. It is thus necessary to quantify how the uncertainty of the cross-section values is propagated through the model for the neutron flux. High-dimensional model representations (HDMRs) can be employed to systematically quantify input-output relations. It can, however, be computationally prohibitive to construct a surrogate model using the HDMR framework for a model that has thousands of parameters. In this paper, we introduce an algorithm that utilizes the New Morris Method to first reduce the parameter space to include only the significant individual and pairwise effects and then construct a surrogate model using a Cut-HDMR expansion within the reduced space. A unified index is introduced to facilitate the comparison of the significance of the model parameters. The accuracy and efficiency of the surrogate model is demonstrated using a one-dimensional neutron transport equation.}, number={3}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hu, Zhengzheng and Smith, Ralph C. and Willert, Jeffrey and Kelley, C. T.}, year={2014}, month={Jul}, pages={350–360} }
 @article{baskaran_hu_lowengrub_wang_wise_zhou_2013, title={Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation}, volume={250}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2013.04.024}, abstractNote={In this paper we present two unconditionally energy stable finite difference schemes for the modified phase field crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic phase field crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Baskaran, Arvind and Hu, Zhengzheng and Lowengrub, John S. and Wang, Cheng and Wise, Steven M. and Zhou, Peng}, year={2013}, month={Oct}, pages={270–292} }
 @inproceedings{hu_smith_burch_hays_oates_2013, title={Homogenized energy model and markov chain Monte Carlo simulations for macro fiber composites operating in broadband regimes}, booktitle={Proceedings of the ASME Conference on Smart Materials, Adaptive Structures and Intelligent Systems, vol 1}, author={Hu, Z. Z. and Smith, R. C. and Burch, N. and Hays, M. and Oates, W. S.}, year={2013}, pages={321–327} }
 @inproceedings{hu_smith_2012, title={A strain model for piezoelectric materials operating in highly hysteretic regimes}, DOI={10.1115/smasis2011-5073}, abstractNote={Piezoelectric materials exhibit hysteresis in the field-strain relation at essentially all drive levels. Furthermore, this non-linear relation is dependent upon both prestresses and dynamic stresses generated during employment of the materials. The accurate characterization of this nonlinear and hysteretic material behavior is critical for material characterization, device design, and model-based control design. In this paper, we will discuss the characterization of hysteresis using the homogenized energy model (HEM) framework. At the mesoscale, energy relations characterizing field and stress-dependent 90 and 180 degree switching are used to develop fundamental kernels or hysterons. Material and field nonhomogeneities are subsequently incorporated by assuming that certain parameters are manifestations of underlying densities. This yields a macroscopic model that accurately characterizes the fundamental material behavior yet is sufficiently efficient for optimization and control implementation. Attributes of the model will be illustrated through comparison to experimental data.}, booktitle={Proceedings of the ASME Conference on Smart Materials, Adaptive Structures and Intelligent systems (SMASIS 2011), vol 2}, author={Hu, Z. Z. and Smith, Ralph}, year={2012}, pages={241–249} }
 @article{hu_smith_ernstberger_2012, title={Data-driven techniques to estimate parameters in a rate-dependent ferromagnetic hysteresis model}, volume={407}, ISSN={["1873-2135"]}, DOI={10.1016/j.physb.2011.06.084}, abstractNote={Abstract The quantification of rate-dependent ferromagnetic hysteresis is important in a range of applications including high speed milling using Terfenol-D actuators. There exist a variety of frameworks for characterizing rate-dependent hysteresis including the magnetic model in Ref. [2] , the homogenized energy framework, Preisach formulations that accommodate after-effects, and Prandtl–Ishlinskii models. A critical issue when using any of these models to characterize physical devices concerns the efficient estimation of model parameters through least squares data fits. A crux of this issue is the determination of initial parameter estimates based on easily measured attributes of the data. In this paper, we present data-driven techniques to efficiently and robustly estimate parameters in the homogenized energy model. This framework was chosen due to its physical basis and its applicability to ferroelectric, ferromagnetic and ferroelastic materials.}, number={9}, journal={PHYSICA B-CONDENSED MATTER}, author={Hu, Zhengzheng and Smith, Ralph C. and Ernstberger, Jon M.}, year={2012}, month={May}, pages={1394–1398} }
 @article{smith_hu_2012, title={Homogenized energy model for characterizing polarization and strains in hysteretic ferroelectric materials: Material properties and uniaxial model development}, volume={23}, ISSN={["1530-8138"]}, DOI={10.1177/1045389x12453967}, abstractNote={ Ferroelectric materials, such as lead zirconate titanate, lanthanum-doped lead zirconate titanate, and BaTiO3, are being considered, or are already being employed, for a large number of applications including nanopositioning, high-speed valves for fuel injectors, ultrasonic transducers, high-speed camera shutters and autofocusing mechanisms, energy harvesting, and pico air vehicle design. Their advantages include nanometer positioning resolution, broadband frequency responses, moderate power requirements, the capability for miniaturization, and complementary actuator and sensor capabilities. However, they also exhibit creep, rate-dependent hysteresis, and constitutive nonlinearities at essentially all drive levels due to their noncentrosymmetric nature. In this article, we model the hysteretic dependence of strains and polarizations on input fields and stresses using the homogenized energy model framework. At the domain level, the minimization of the Gibbs energy densities yields linear constitutive relations. Nonlinearities and hysteresis due to dipole switching are modeled at the grain level using the Boltzmann theory to specify the evolution of dipole fractions that serve as internal variables. In the final step of the development, stochastic homogenization, based on the assumption that interaction fields and driving forces are manifestations of underlying densities, is used to construct nonlinear constitutive relations for the bulk material. It is demonstrated that these relations are amenable to subsequent development of distributed system models. The article includes significant discussion regarding the mechanisms that produce hysteresis in ferroelectric materials. The capability of the framework for characterizing various hysteretic phenomena, including creep and various rate dependencies, is illustrated by validation with lead zirconate titanate and lanthanum-doped lead zirconate titanate data. }, number={16}, journal={JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES}, author={Smith, Ralph C. and Hu, Zhengzheng}, year={2012}, month={Nov}, pages={1833–1867} }
 @article{hu_lowengrub_wise_voigt_2012, title={Phase-field modeling of epitaxial growth: Applications to step trains and island dynamics}, volume={241}, ISSN={["1872-8022"]}, DOI={10.1016/j.physd.2011.09.004}, abstractNote={Abstract In this paper, we present a new phase-field model including combined effects of edge diffusion, the Ehrlich–Schwoebel barrier, deposition and desorption to simulate epitaxial growth. A new free energy function together with a correction to the initial phase variable profile is used to efficiently capture the morphological evolution when a large deposition flux is imposed. A formal matched asymptotic analysis is performed to show the reduction of the phase-field model to the classical sharp interface Burton–Cabrera–Frank model for step flow when the interfacial thickness vanishes. The phase-field model is solved by a semi-implicit finite difference scheme, and adaptive block-structured Cartesian meshes are used to dramatically increase the efficiency of the solver. The numerical scheme is used to investigate the evolution of perturbed circularly shaped small islands. The effect of edge diffusion is investigated together with the Ehrlich–Schwoebel barrier. We also investigate the linear and nonlinear regimes of a step meandering instability. We reproduce the predicted scaling law for the growth of the meander amplitude, which was based on an analysis of a long wavelength regime. New nonlinear behavior is observed when the meander wavelength is comparable to the terrace width. In particular, a previously unobserved regime of coarsening dynamics is found to occur when the meander wavelength is comparable to the terrace width.}, number={2}, journal={PHYSICA D-NONLINEAR PHENOMENA}, author={Hu, Zhengzheng and Lowengrub, John S. and Wise, Steven M. and Voigt, Axel}, year={2012}, month={Jan}, pages={77–94} }
 @article{spayd_shearer_hu_2012, title={Stability of plane waves in two-phase porous media flow}, volume={91}, ISSN={["0003-6811"]}, DOI={10.1080/00036811.2011.618128}, abstractNote={We examine the Saffman–Taylor instability for oil displaced by water in a porous medium. The model equations are based on Darcy's law for two-phase flow, with dependent variables pressure and saturation. Stability of plane wave solutions is governed by the hyperbolic/elliptic system obtained by ignoring capillary pressure, which adds diffusion to the hyperbolic equation. Interestingly, the growth rate of perturbations of unstable waves is linear in the wave number to leading order, whereas a naive analysis would indicate quadratic dependence. This gives a sharp boundary in the state space of upstream and downstream saturations separating stable from unstable waves. The role of this boundary, derived from the linearized hyperbolic/elliptic system, is verified by numerical simulations of the full nonlinear parabolic/elliptic equations.}, number={2}, journal={APPLICABLE ANALYSIS}, author={Spayd, Kim and Shearer, Michael and Hu, Zhengzheng}, year={2012}, pages={295–308} }
 @article{hu_smith_ernstberger_2012, title={The homogenized energy model for characterizing polarization and strains in hysteretic ferroelectric materials: Implementation algorithms and data-driven parameter estimation techniques}, volume={23}, ISSN={["1530-8138"]}, DOI={10.1177/1045389x12453968}, abstractNote={ Ferroelectric materials, such as lead zirconate titanate, lanthanum-doped lead zirconate titanate, lead manganese niobate, and BaTiO3, provide unique actuator and sensor capabilities for applications including nanopositioning, high-speed valves and fuel injectors, camera focusing and shutter mechanisms, ultrasonic devices for biomedical imaging and treatment, and energy harvesting devices. However, to achieve the full potential of the materials, it is necessary to develop and employ models that quantify the creep, rate-dependent hysteresis, and constitutive nonlinearities that are intrinsic to the materials due to their domain structure. The success of models requires that they be highly efficient to implement since real-time applications can require kilo hertz to mega hertz rates. The calibration of models for specific materials, devices, and applications requires efficient and robust parameter estimation algorithms. Finally, control designs can be facilitated by models that admit efficient and robust approximate inversion. The homogenized energy model is a multiscale, micromechanical framework that quantifies a range of hysteretic phenomena intrinsic to ferroelectric, ferromagnetic, and ferroelastic materials. In this article, we present highly efficient implementation and parameter estimation algorithms for the ferroelectric model. This includes techniques to construct analytic Jacobians and data-driven algorithms to determine initial parameter estimates to facilitate subsequent optimization. The efficiency of these algorithms facilitates material and device characterization and provides the basis for constructing efficient and robust inverse algorithms for model-based control design. The model implementation, calibration, and validation are illustrated using rate-dependent lead zirconate titanate data and single-crystal BaTiO3 data. }, number={16}, journal={JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES}, author={Hu, Zhengzheng and Smith, Ralph C. and Ernstberger, Jon}, year={2012}, month={Nov}, pages={1869–1894} }
 @article{hu_haider_2011, title={Algebraic Multigrid Preconditioning for Finite Element Solution of Inhomogeneous Elastic Inclusion Problems in Articular Cartilage}, volume={3}, ISSN={["2070-0733"]}, DOI={10.4208/aamm.10-m1070}, abstractNote={Abstract}, number={6}, journal={ADVANCES IN APPLIED MATHEMATICS AND MECHANICS}, author={Hu, Zhengzheng and Haider, Mansoor A.}, year={2011}, month={Dec}, pages={729–744} }
 @article{hu_smith_stuebner_hays_oates_2011, title={Statistical Parameter Estimation for Macro Fiber Composite Actuators using the Homogenized Energy Model}, volume={7978}, ISSN={["0277-786X"]}, DOI={10.1117/12.884622}, abstractNote={Macro Fiber Composites (MFC) are planar actuators comprised of PZT fibers embedded in an epoxy matrix that is sandwiched between electrodes. Due to their construction, they exhibit significant durability and flexibility in addition to being lightweight and providing broadband inputs. They are presently being considered for a range of applications including positioning and control of membrane mirrors and configurable aerospace structures. However, they also exhibit hysteresis and constitutive nonlinearities that must be incorporated in models to achieve the full potential of the devices. In this paper, we discuss the development of a model that quantifies the hysteresis and constitutive nonlinearities in a manner that promotes subsequent control design. The constitutive model is constructed using the homogenized energy framework for ferroelectric hysteresis and used to develop resulting system models. The performance of the models is validated with experimental data.}, journal={BEHAVIOR AND MECHANICS OF MULTIFUNCTIONAL MATERIALS AND COMPOSITES 2011}, author={Hu, Zhengzheng and Smith, Ralph C. and Stuebner, Michael and Hays, Michael and Oates, William S.}, year={2011} }