A human being is part of the whole called by us universe,
a part limited in time and space. ...
We shall require a substantially new manner of thinking
if humanity is to survive.
(Albert Einstein , 1879-1955)

Sustainable development, mathematics in industry, human health and environment

As it stands now, humanity depletes its resources with an astronomical speed, much faster than they can be replenished. It is important to understand that the only way for our further development could be a balance between our needs and preserving the environment. We are interested in the development of mathematical foundations for and mathematical modeling of alternative and renewable sources of energy. Such sources should help reducing the damage to the environment caused by more traditional sources of energy and fuels. This includes, for example, solar energy. We are also interested in the development of biological and chemical sensors that, along with other applications, would allow better environmental monitoring. Although small particles originating from human activities have been around since the dawn of our civilization, our technological development has dramatically increased the variety of such particle chemical compositions. With the advance of nanotechnology, we have to be aware of the effects such particles may produce. These and other issues of health and environments and their relationship to modern technologies are part of our interests in these focus area.

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

• Mathematical models for renewable and alternative sources of energy;
• Industrial & engineering applications of mathematics;
• Mathematical models for medical and environmental sciences.

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Mathematical models for renewable and alternative sources of energy

Furthermore, we note that by using quantum dot arrays [ EJ-79 ], a very high efficiency of solar-electric conversion can be archived. In particular, harvesting the maximum proportion of the incident light can be archived if we would use an array of different size quantum dots. We have analyzed  size and shape effects in quantum dots and other low dimensional nanostructures in a series of our publications [ EJ-62 , EJ-65 , EJ-73 ] (also proceedings [ EP-25 , Cambridge-2010 ]).

It is also known that ZnO based low dimensional nanostructures can also be used in solar cells and we have analyzed these materials with density functional theory and nonequilibrium Green's function technique in [ EJ-125 , EJ-139 ].

We have analyzed systems for energy transfer (power generation/refrigeration) via the Peltier effects in the context of thermoelectric materials [ EJ-42 , EJ-74 ], the materials which we explored also in the context of low dimensional nanostructures [ EJ-138 ]. Currently, thermoelectrics are being explored as an important element of solar thermal energy generation systems and similar systems for alternative sources of energy.

Our other contributions include the development of nonlinear models for the description of dynamic properties of electric vehicles [ EJ-54 ], as well as mathematical models for an environmental friendly methodology for the implementation of heat recovery and airflow control systems [ EJ-87 , EJ-102 ].

For more information on mathematical models for quantum dots, please see here.

Industrial & engineering applications of mathematics

In this field, our research is closely related to the following technologies:

• nanotechnology and bio-nanotechnology;
• sensors and actuators technology; smart materials and structures technology; nondestructive evaluation technology;
• technologies relying on interactions of structures with gas/fluids and on solving multi-physics, multiscale problems.

A fairly detailed account of our research in the field of nanotechnology can be found here. In many ways, the class of problems in bio-nanotechnology we are interested in has been motivated by our interest in multiscale mathematical problems in biology, medicine, and nanotechnology. Details of this are given in the next section and here.

Sensors and actuators are key elements for humans and for human-made systems to interact with our environment. Many sensors and actuators are designed based on smart materials and in this sense these two technologies are not independent. Furthermore, due to a trend of reducing the size of sensors and actuators, in many applications nanotechnology plays an increasingly important role in this field. These technologies (separately or in their combination) have opened opportunities for nondestructive testing which includes a range of methodologies from ultrasonic to electromagnetic and to optic methods.  In a nutshell (and apart from nanotechnological issues which are discussed separately here), our interest in this diverse field focuses on piezoelectric and shape memory materials (SMA) and devices, but include also other smart (intelligent, adaptive) materials such as electroactive polymers and ferroelectrics, as well as thermoelectrics and other materials and devices based on these materials. Some of the publications on thermoelectric SMA actuators and shape memory alloys in engineering and industrial context can be found in [ EJ-42 ] and [ EJ-69 , EJ-91 , EJ-92 , EJ-100 , EJ-111 , EJ-114 , EJ-117 ], respectively (with a more detailed account on these and other coupled problems found here and in a number of special journal issues, including [ EJ-68 , EJ-84 , EJ-88 , EJ-97 , EJ-113 , EJ-121 , EJ-124 ]).

Thermal analysis of complex systems in industrial applications, developing mathematical models and numerical discretization for such an analysis, has been a topic of continuing interest, covering nonlinear mathematical models for thermal analysis of integrated circuits in semiconductor industry [ EJ-12 , EJ-16 ] as well as mathematical models for thermal analysis of electric induction motors [ EJ-52 ]. We also developed a new model and numerical approximation to analyze the effect of the presence of molecular water impurity of various concentrations in absorbing and scattering glass media on the temperature field in a layer subjected to thermal infrared radiation [ EJ-77 ]. Our contributions to the development of efficient numerical procedures for energy balance models are found in [ EJ-20 , EJ-30 , EJ-32 , EJ-37 , EJ-51 , EJ-70 ] . Such models are able to account for non-local, non-equilibrium processes and have a wide range of applicability. We demonstrated the efficiency of the developed models and their numerical approximations in describing important phenomena in semiconductor plasma, including velocity overshoot, fast photo-response of optically sensitive semiconductors for photonics applications and others. Our contributions to modelling complex systems for renewable and alternative sources of energy can be found here, while more details on applied coupled problems of industrial interest, including thermoelasticity, electroelasticity and others, can be found here.

The dynamic behaviour of ferroelectric materials and multilayer structures is an important component of many current and potential applications. Mathematically, associated mathematical models are strongly nonlinear and represent substantial challenges for the analysis and the development of efficient numerical techniques. By symmetry considerations, most ferroelectric materials are also piezoelectric and pyroelectric which make them very useful as capacitors in sensor applications, among many others. We developed an efficient model for the analysis of these materials, as well as numerical procedures to carry out such an analysis, and some cases their control [ EJ-127 , EJ-132 , EJ-141].

The analysis of thermo-piezoelectric coupling was carried out in [ EJ-39 , EJ-56 ] and the full coupling between a piezoelectric structure and gas/fluid media has been analyzed by us in [ EJ-50 , EJ-85 ] where we also discussed the solution methodology for such coupled problems.

The importance of and trends in interdisciplinary education were discussed in our contribution [ EJ-57 ] where we proposed core curriculum subjects for university departments specializing in or with a strong component of applied mathematics and mathematical modelling, reflecting interdisciplinary challenges that must be part of new educational practices .

Several special journals issues appeared where many challenging problems of industrial and engineering mathematics can be found along with state-of-the-art tools for their solutions [ EJ-68 , EJ-88 , EJ-97 , EJ-113 , EJ-121 , EJ-124 ].

Mathematical models for medical and environmental sciences

We analyzed several systems for drug delivery [ EJ-47 ] and looked into novel potential applications of RNA nanostructures in medical sciences, focusing on studying their properties [ EJ-137 , EJ-145 ].

We contributed to the development of mathematical models and efficient numerical methods for dynamic problems in bio-inspired materials such as shape memory alloys (SMAs). These materials find a wide range of applications in orthodontics and dental biomechanics , optometry, orthopedic surgery, biologically-friendly replacements in body cavities, arteries and other vascular applications, implants, and in various surgery instruments. While there exists a vast amount of literature on constitutive and time-independent modelling of SMAs and SMA-based systems, our main focus over the years has been of the dynamics. This is quite important feature in biological and medical applications of SMAs. Two key technologies are related to this: smart materials and structures technology and nondestructive evaluation technology. The latter includes the development of non-invasive (or less invasive) medical diagnostics. We first proposed 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 in [ EJ-93 ] for both uncoupled and coupled problems). This allowed to treat both stress-induced and temperature-induced phase transformations in these materials and related hysteresis phenomena in a unified manner [ EJ-44 ], with further generalization to include hyperbolicity of heat conduction to follow [ EJ-49 ]. 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 ]. A new hybrid optimization procedure was developed and applied in [ EJ-118 ]. 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 ]. We have also developed several efficient procedures for model reductions in this field based on the manifold reduction, the Proper Orthogonal Decomposition (POD), and the Galerkin projection techniques [ EJ-33 , EJ-55 , EJ-106 , EJ-141 ]. More details on our research into the dynamics of SMAs and SMA-based devices can be found here.

Polymers and composites are another class of materials that find numerous important applications in medical and biological science. Some of our contributions to this field can be found in [ EJ-58 , EJ-78 , EJ-105 , EJ-109 , EJ-126 ]. We note that that in polymeric materials applied in medical applications there are many non-trivial phenomena that need to be understood much better. We initiated the analysis of one such phenomena known as thermal spiking. We developed a new model for this phenomenon, as well as an efficient method for its numerical solution, and presented results on thermal spiking on an example of bone cement [ EJ-45 , EJ-59]. We showed that mathematically such problems are reducible to a coupled system of ODEs and PDE. Other authors used such mixed coupled models for other biological and medical applications, e.g. in vascular remodelling in biological tissues. The techniques we developed may proved to be quite useful in those areas too.

In the context of environmental sciences, we analyzed models for climate systems [ EJ-23 ] (also proceedings [ EP-3 ]). Climate belongs to the class of systems whose dynamics are only observable in transient states and the resulting mathematical models are characterized by the fact that small-scale phenomena influence the large-scale properties of the modelling system (with the former that cannot be extracted from the latter using available hardware and computational procedures). We provided arguments for such models to be treated as a link between coupled physical processes and computational decoupling with subsequent implications to further climate predictions.

Our contributions to the development of mathematical models for renewable and alternative sources of energy are described here.

We are interested in the development of mathematical models for biological and chemical sensors (including those can be used in harsh environments for environmental monitoring), as well as for the field emission as an alternative to the conventional X-ray radiation. In particular, we proposed a series of new multiscale multi-physics models for the evolution of Carbon Nanotubes (CNTs) on thin films, numerical methods for the solution of the resulting problems, and we reported the efficiency of the proposed models and techniques based on theoretical, computational, and experimental results [ EJ-112 , EJ-116 , EJ-122 , EJ-131 ].

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.

Piezoeffect in biological tissues has been a subject of long standing interest and controversies. We analyzed this effect in a dynamic setting accounting for the hydration effect in human cornea [ EJ-99 ]. Other problems of piezoelectricity, including the issues of existence, uniqueness, and regularity of solutions for the corresponding time-dependent mathematical models can be found here.

Several models for the analysis of relaxation of DNA molecules were proposed in [ EJ-105 , EJ-109 ] and more information about our research in these areas can be found here. These include also RNA silencing phenomena [ EP-67 ], modelling genetic networks as a multiscale process [ EP-37 ], and the analysis of cell cycle dynamics [ EJ -130 ].

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