   The whole is more than
the sum of its parts.

Multiscale coupled systems and phenomena:

analysis and applications

Multiscale coupled systems are ubiquitous in sciences, engineering, and society. All phenomena we observe in Nature are reflections of various forms couplings, e.g. between different physical fields, between different components of a system, or between different systems. A key to our better understanding of such multiscale coupled systems, effects, and phenomena is kept in their rigorous mathematical description. At the same time, due to coupling and multiscale characteristics of these systems, effects, and phenomena, such a description, consisting in the development of adequate mathematical models and methodologies for their solutions, leads to some of the most challenging problems of modern science & engineering.

In this focus area, our research interests have included the following topics:

Coupled fields theory and its applications: electro-mechanical fields, thermo-elasticity, thermo-electricity, thermo-electroelasticity, magneto-thermo-electroelasticity

Some of the most important and challenging problems in science and engineering are coupled problems. The coupling is pronounced at three basic levels: (i) the level of field coupling as it is the case in thermo-elasticity, electro-elasticity and in many other areas of applications, (ii) the level of coupled effects between different parts of the system, and/or (iii) the level of coupling between different spatial/spatio-temporal scales at which the natural or man-made system, that is being studied, operates. This coupling brings to life new phenomena, the phenomena which are not amenable to an adequate description with conventional mathematical models, traditionally developed without accounting for the coupled nature of the problem. An early observation that the coupling in nature plays a profound role was made by Aristotle who noted that "the whole is more than the sum of its parts." The development of coupled mathematical models has an interesting history with one of the first examples attributed to Pierre Simon Laplace who corrected Newton's purely mechanical theory of sound propagation by accounting for thermal field effects. Another important example includes the development of a dynamic model of coupled electro-magnetic interactions by James Clerk Maxwell. At the M²NeT Lab, we focus on the development of coupled mathematical models for addressing fundamental questions of science and solving new problems in science and engineering. We study coupled phenomena, effects, and processes that are becoming increasingly important in the areas of nano- and bio-nano technologies, in new technologies based on smart materials and structures, biocompatible materials and bio-inspired systems, and in other areas.

Dynamic Thermoelasticity (linear and nonlinear models).

As noted, early attempts of creating a theory that would describe mechanical and thermal fields as a unified process go back to P. S. Laplace with his ideas of correcting Newton's purely mechanical theory of sound propagation. However, it is only with the advent of the applications where thermal stresses were becoming increasingly important, the dynamic theory of thermoelasticity was put firmly on the map of research in mathematics and mechanics. In particular, an important modern precursor for the development of coupled dynamic thermoelasticity was laid down by the works of V. I. Danilovskaya who studied thermal stress waves generated by boundary in the early 1950-ies. Most of the works in the field were related to the standard displacement-temperature formulation. We noted that in many applied problems it is the stress components, along with the temperature, are the main unknowns to be determined. As a result, we proposed to solve the problem of coupled dynamic thermoelasticity in the stress-temperature formulation [ EJ-8 , EJ-17 , EJ-41 ]. Based on the Steklov's operator technique combined with the application of the Bramble-Hilbert lemma, we obtained a priori estimates that were dependent on the solution regularity. A general family of operator-difference schemes for the solution of the underlying problems was derived and convergence of such schemes was analyzed in classes of generalized (weak) solutions. Furthermore, a detailed dispersion analysis for such fully coupled problems was carried out and it was shown that the stability conditions for derived schemes can be obtained with the Cayley transform technique [ EJ-41 ].

We have also been contributing to nonlinear coupled dynamic thermoelasticity. Our main interests here were centered around problems describing multiphase solids such as materials with shape memory and phase transformations. In particular, in order to describe low (martensite) and high (austenite) temperature phases in such materials simultaneously and transformations between these phases, we were using the Landau-Devonshire-Ginzbrurg representation of the free energy function, leading to a quintic nonlinearity in the stress-strain constitutive models. Due to this high order nonlinearity and intrinsic coupling between the equations, the resulting problems are at the forefront of challenges in modern applied and computational mathematics. Our works in this field can be divided into a few groups, (A) to (D).

(A) - Model Reduction. We were interested in the analysis of general three-dimensional (3D) models describing phase transformations, focusing on the development efficient procedures for model reductions. We were the first to propose a reduction of the full 3D PDE-based models for materials with shape memory to a more tractable differential-algebraic systems [ EJ-33 ] (further discussed and exemplified in [EJ-93 ] for both uncoupled and coupled problems, with additional examples for SMA patches given in [ EJ-86 ]). This original procedure was based on the center manifold theory. This followed immediately by applications of the reduced model where we were able to treat both stress-induced and temperature-induced phase transformations in these materials and related hysteresis phenomena in a unified manner [ EJ-44 ]. A generalization of the original model and to include hyperbolic heat conduction effects was also obtained and discussed in [ EJ-49 ]. This generalized model can be also treated with the developed model reduction technique in a straightforward manner. Details of the developed theory for center-manifold-based model reduction of general 3D models for phase transformation in the context of materials with memory were provided in [ EJ-55 ]. Our model reduction procedures based on the Proper Orthogonal Decomposition (POD) and the Galerkin projection techniques can be found in [ EJ-106 , EJ-141 ].

(B) - Development of Efficient Numerical Methods. For the first time, conservative numerical approximations in this field of dynamic problems were developed and justified mathematically in [ EJ-53 , EJ-72 ]. Chebyshev's collocation and Chebyshev's spectral procedures were developed for these problems in [ EJ-100 , EJ-107 ] (also proceedings [ Cruz-2009 ]) which also included the Rayleigh dissipation term [ EJ-111 ]. We developed a Finite Element Method (FEM) based approach that incorporates the lattice kinetics, involving the order variables, and non-equilibrium thermodynamics [ EJ-114 ]. Its variants, including those applied to 3D problems, were discussed in detail in [ EJ-82 , EJ-92 , EJ-104 ] and our new Finite Volume Method procedure for these dynamic problems was reported in [ EJ-101 ].

(C) - Applications. The development of models and their efficient numerical solutions in this field having in mind specific physical or engineering systems has been an important part of our research too. Materials with shape memory have applications from aerospace engineering to biomechanics and from medical practices to low dimensional nanostructures. We discussed several applications of these materials in engineering and industrial contexts, where they often have to be treated as multiscale coupled dynamic problems [ EJ-74 ], in [ EJ-42 , EJ-69 , EJ-91 , EJ-92 , EJ-100 , EJ-107 ,  EJ-111 EJ-114 , EJ-117]. More on biological and medical applications of materials with memory can be found here.

(D) - Optimization and Control. A new hybrid optimization procedure for the models describing phase transformations in materials with memory was developed and applied in [ EJ-118 ]. The methodology proposed was based on an initial estimate of the global solution by a genetic algorithm, followed by a refined quasi-Newton procedure to locally refine the optimum. By combining the local and global search algorithms, the developed optimization procedure was demonstrated on several examples. To control such multiphase complex systems as materials with shape memory is a difficult task. This task was undertaken in our paper [ EJ-132 ] in a closely related context of ferroelectric materials. Theoretical issues of complex systems control were discussed in [  EJ-21 , EJ-60 , EJ-129 ].

Most recent applications of coupling between thermal and elastic fields have been explored by us in the context of low dimensional nanostructures [ EJ-135 , EJ-138 ].

Dynamic Electroelasticity and Electromechanical Fields (piezoelectric effects, linear and nonlinear models).

Practical applications of electroelasticity theory started at least from the discovery  in 1880 by the brothers Pierre and Jacques Curie the direct piezoelectric effect, followed a year later by the works of Gabriel Lippmann on the converse piezoelectric effect. This effect is pronounced by the ability of certain materials to generate an electric field in response to applied mechanical strain. The applicability of such materials is surprisingly large, including a multitude of physical and engineering systems (where crystals, certain ceramics, or piezoelectric polymers are in the heart of their functioning), as well as biological systems, ranging from bones and biological tissues to DNA and proteins.

The original development of mathematical models in the field of piezoelectricity is largely due to Woldemar Voight and his book on crystal physics published in 1910. Despite many applications of dynamic theory of piezoelectricity, the well-posedness and regularity of solutions of underlying mathematical equations remained open until the late 1980ies. We were the first who succeeded to provide a rigorous mathematical proof of such coupled dynamic models' well-posedness and who analyzed the regularity of their solutions [ EJ-3 , EJ-5 , EJ-7 ]. The main summary of the results are given in [ EJ-11 , EJ-22 , EJ-24 ] and [ EJ-31 ] for one- and multi-dimensional cases, respectively. Efficient variational-difference numerical approximations were also developed at that time with a hierarchy of a priori estimates dependent of the regularity of the underlying solution [ EJ-1 , EJ-2 , EJ-6 , EJ-13 , EJ-19 ]. Those early results were the basis for the development a family of variational-difference schemes for the solution of coupled dynamic problems of electroelasticity describing piezoelectric phenomena, a generalization of the Courant-Friedrichs-Lewy stability condition to the case of coupled electroelasticity, obtained by us for the first time, as well as for obtaining  a scale of accuracy results for generalized solutions of such problems and convergence theorems [ EJ-22 , EJ-24 , EJ-26 , EJ-29 , EJ-31 , EJ-71 ] . This provided a rigorous mathematical justification for the application of dynamic models coupling electric and mechanical fields and capable of describing piezoelectric phenomena.

As a result, our later works in this field were focused on the application of such models to important problems in the analysis and modelling of biological systems (e.g., [ EJ-99 ]), as well as of low dimensional nanostructures such as quantum nanowires, dots, carbon nanotubes (e.g., [ EJ-90 , EJ-116 , EJ-135 , EJ-136 , EJ-138 , EJ-146 ]), where coupled electromechanical fields are of crucial importance.

Thermoelectricity.

The essence of thermoelctricity is a coupling between temperature differences and electric voltage. There are several important effects where thermoelectricity plays a key role, namely the Seebeck, the Peltier effect, and the Thomson effects. These thermoelectric effects have a wide range of applications starting from their traditional cooling applications to their applications in nanotechnology. Our original interest in thermoelectricity was due to modelling issues of SMA actuators [ EJ-42 ]. Many problems arising in this context lead to multiscale coupled dynamic models [ EJ-74 ] and we have continuing interest in developing efficient numerical methods for dealing with such models. In particular, our current interest in the topic is due to the importance of thermoelectricity in modelling low dimensional nanostructures [ EJ-135 , EJ-138 ].

Ferroelectricity and Flexoelectricity.

In many materials we observe a spontaneous electric polarization that can be reversed by the application of an external electric field. This phenomenon is known as ferroelectricity. To study this phenomenon is difficult due to the fact that its characteristics are dependent on a range of different factors, including dimensions of the material sample, temperature, surface effects and boundary conditions, etc. Landau theory provides a foundation upon which the mathematical modelling of ferroelectric materials is based. Corresponding (thermodynamic) mathematical models, derived from the expansion of the free energy function, serve as a bridge between microscopic and macroscopic modelling frameworks. Due to contributions of Devonshire specifically in the context of ferroelectric and Ginzburg in the context of finite size ferroelectrics, the resulting models are usually termed as the models of Landau-Devonshire-Ginzburg type. It is these models that represent our main interest in this field.

Ferroelectricity is becoming increasingly important at the nanoscale due to its size dependence. We are interested in this and other similar effects at the nanoscale such as flexoelectricity where the material may exhibit a spontaneous electrical polarization induced by a strain gradient. In fact, recent studies show that low dimensional nanostructures may exhibit nonlocal electromechanical effects such as flexoelectricity. At the same time, despite the recent progress in nanostructure growth techniques imperfections such as defects are practically unavoidable. These imperfections may enhance flexoelectric effects. We analyzed the effects of these imperfections on linear electromechanical properties and on flexoelectricity, focusing on the study of GaN-based quantum dots embedded in an AlN matrix [ EP-79 ].

Coupled Dynamic Thermo-electroelasticity, Magneto-thermo-electroelasticity, Other Coupled Effects.

Mathematical models of thermo-electroelasticity and numerical methods for their solutions were studied in [ EJ-39 , EJ-56 ]. The interactions of piezoelectric structures with fluid media as a representative example of fluid-structure interaction problems were studied by us in [ EJ-85 ]. On a closely related note we have also been interested in interactions of electromechanical fields with acoustic media. Such problems are important in many applications in the classical mechanics setting [ EJ-50 ], as well as in the context of modelling opto-electronic media [ EJ-14 ] where quantum effects may become important. The latter is of interest to us in the context of low dimensional semiconductor nanostructures where we also analyzed systematically the influence of thermo-electromechanical fields on the properties of such structures [ EJ-135 , EJ-138 ], as well as the influence of the magnetic field as a control parameter [ EJ-146 ].

Waves in complex media, highly oscillatory phenomena, coupling continuum and quantum worlds

We are interested waves in microstructures and associated multiscale phenomena. This interest has been largely motivated, but not limited to, the analysis of microstructures in multiphase materials such as materials with shape memory, described in detail in the previous section.

Other topics of interest here include wave phenomena in polymers and composites (e.g., thermal spiking [ EJ-45 , EJ-59]), in layered media [ EJ-18 ] (e.g., semiconductor-piezoelectric layers [ EJ-14 ] and nanolayers [ EP-82 ]), in DNA related processes [ EJ-105 , EJ-109 , EJ-126 ], and other physical, biological, and engineering systems.

Many problems in these areas require dealing with highly oscillatory functions and their integration. In addition, in many applications we have to account for the fact that the information about such functions come from experiments and usually is incomplete. Optimal and other efficient formulae for integrating such functions in one- and multi-dimensional cases were derived in a series of our papers [ EJ-27 , EJ-43 , EJ-48 , EJ-66 ]. We are also interested in the spectral analysis of complex systems and associated eigenvalue problems [ EJ-34 ], as well as in fundamental and applied problems on the border between classical and quantum mechanics [ EJ-133 , EJ-138 ].

Coupled models in Molecular Dynamics and First Principles calculations, lattice models, phase transformations and critical phenomena

In a subset of coupled multiscale problems we are dealing with, we need to get a more refined insight into some properties of the structures or materials of interest. In this case, we develop Molecular Dynamics and First Principles based methodologies. Examples of our endeavours in this field include Nonequilibrium Molecular Dynamics (NMD) studies of nanowires [ EJ-144 ] and Molecular Dynamics studies of RNA nanostructures [ EJ-137 ] with subsequent development of a hierarchy of models for carrying out systematic coarse-graining of such structures [ EJ-145 ]. First Principles methodologies were applied to our comprehensive studies of aluminum-nickel and calcium compound systems [EJ-134 , EJ-140 ], while our newly developed combined nonequilibrium Green's function and density functional theory methodology was applied to studies of zinc-oxide nanowires [ EJ-139 ] (with nickel-aluminum nanolayers studied by us in [ EP-82 ] and relative stability of nanosized zinc-oxide studied in [ EJ-125 ]).

High performance computing is also an essential element in our studies of lattice spin models where we focus on the behaviour near the critical (phase transition) point by using non-perturbative approaches based on Monte-Carlo methods [ EJ-103 , EJ-115 , EJ-143 ]. This includes our analysis of Goldstone singularity in XY models and the Goldstone mode effects in three-dimensional O(4) models. The Wolff cluster algorithm provides one of the most effective tools to simulate important properties of many particle systems near the critical point. However, a major drawback in the application of this algorithm were related to difficulties in its parallelization, in particular in the application to large lattices. On the example of three-dimensional Ising models, in [ EJ-142 ] we demonstrated for the first time how parallelization of the Wolff single cluster algorithm can be achieved. Our procedure is generalizable to other lattice spin models and provides an important new tool for the simulation of very large lattices near the critical phase transition point.

Phase transformations are universal phenomena occurring frequently in physics and chemistry, biology and medicine, engineering and finance. The procedures developed by us here are potentially useful in other applications ranging from quantum spin problems to financial applications (see also here) and to Barabasi-Albert networks [ EJ-142 ].

Conservative numerical schemes and variational difference approaches; numerical analysis; model reductions; well-posedness and regularities of coupled PDEs

Our results in this area pertain to coupled systems of PDEs, including hyperbolic-parabolic and hyperbolic-elliptic systems. The former arise, for example, in the context of dynamic coupled thermoelasticity and the latter in the context of dynamic coupled electroelasticity. Both these applications have been described in details in this section where details of existence and uniqueness theory for these problems were also given (see summary of theoretical results in [ EJ-11 , EJ-22 , EJ-24 , EJ-31 ]). Closely related topics, also described above, include a generalization of the Courant-Friedrichs-Lewy (CFL) stability condition to such coupled time-dependent systems [ EJ-71 ]. In addition to variational-difference numerical approximations developed by us for such coupled dynamic problems, we the first to derive conservative numerical approximations for strongly coupled nonlinear problems of thermoelasticity such as those arising in the context of Landau-Devonshire-Ginzburg models of structural phase transformations [ EJ-53 , EJ-72 ]. Other methods we developed in this context include Finite Volume Methods (FVM), Finite Element Methods (FEM), and pseudospectral procedures (see details here).

The development of model reduction procedures for coupled nonlinear dynamic problems described by systems of PDEs is another area of interest. Our earlier contributions were in the development of model reductions based on center manifold, Galerkin projection, and Proper Orthogonal Decomposition (POD) techniques [ EJ-33 , EJ-49 , EJ-106 , EJ-141 ].

Dynamics, relaxation time approximations, hyperbolic corrections of parabolic models, extended thermodynamics

We are interested in transport phenomena and dynamic systems that are far from their equilibrium states. Mathematical models in this field can be classified on the basis of relaxation time approximations which in the context of semiconductor modelling of sub-micron structures were applied in  [ EJ-20 , EJ-30 , EJ-32 , EJ-37 , EJ-51 , EJ-70 ]. Theoretical foundations of many such models can be traced to approaches of nonequilibrium thermodynamics. One of such approaches, the extended thermodynamics, has been widely discussed in the literature.

One of the advantages of the above approach lies in the fact that some non-physical phenomena, such as the infinite speed of heat propagation of the processes based on the Fourier law, can be "corrected" within the extended thermodynamics modelling framework. We were analyzing the resulting hyperbolic models in several application areas, including nonlinear models of dynamic coupled thermoelasticity and thermoelastic models for the analysis of polymer dynamics [ EJ-28 , EJ-33 ,  EJ-40 , EJ-49 , EJ-78 ].

We have also been developing methods based on the Nonequilibrium Molecular Dynamics (NMD) as well as tools based on first principles calculations, nonequilibrium Green's function techniques and density functional theory  to study nanostructure properties [EJ-123 , EJ-125 , EJ-134 , EJ-139 , EJ-140 , EJ-144 ].

We are also interested in dynamic problems where the applications of Monte Carlo based procedures can provide an efficient way for their solutions (in the context of phase transformations and financial applications, among others), as well as in modelling complex dynamic biological processes, such as multiscale dynamics of cell cycles [ EJ-130 ].

Coupled effects in complex systems; interactions of light with matter, fluid/gas and structure interactions

Other coupled problems of interest include interactions of light with matter, e.g. with biological systems such as DNA, human tissues, etc. We developed mathematical models and efficient numerical methods for their solutions for studying the effect of thermal damage induced by ultrashort pulsed lasers [ EJ-89 , EJ-96 , EJ-119 ]. Apart from applications in physics, chemistry, and engineering, this research is increasingly important in biology and medicine in the context of preventing thermal damage of human tissues. Other mathematical problems of interest related to human health can be found here.

In addition to biological systems, our interests include also the analysis of multiphase systems and systems with phase transformations under irradiation (ion, UV irradiation, etc). In the context of biological systems recent topics of interests include phototaxis, a movement of an organism in response to the stimulus of light, and the associated hierarchy of mathematical models derived from the Liouville equation for light matter interactions.

Coupled problems pertinent to fluid-structure interactions and other industrial settings have been discussed in this section.

Other applications, including physics, biology, engineering, finance, and environmental science

Environmental sciences provide a wide range of coupled dynamic problems. Our interests here include, but not limited to, the development of mathematical models for more precise chemical and biological sensors, as well as for climate systems models and their error analysis. Details of this and other related issues can be found here. Other problems of interests include neural networks and evolutionary based computations in various applications context [ EJ-120 ] (also proceedings [ EP-T-2010]).

Finally, a wide range of problems we are dealing with require the development of mathematical models for multi-layered, multi-component, or multi-phase structures consisting of several different materials joined together (or components, or phases). While some such problems have already been discussed in the previous section, nanotechnological applications provide a wealth of new challenging problems in this area, from both fundamental and applied perspectives. We are discussing some such problems in the context of low dimensional nanostructures here. 