The numerical computation of the minimum energy for a ferromagnetic material with a fine domain structure modeled by the theory of micromagnetics is difficult because the bulk energy has many local minima and because the optical magnetization can have many domains. We show that an efficient implementation of the simulated annealing algorithm can be used to compute the magnetization which has the minimum energy.
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One feature of shape memory alloys and other smart materials is the formation of fine-scale arrangements (microstructures) as a response to external stimuli. In shape memory materials there is twinning, where the material forms narrow bands in its interior, where in each band the material has a uniform orientation, but in adjacent bands it takes on different but symmetry-related orientations. In this paper, we discuss two different approaches to computing twinned configurations. The first is a variation on the conjugate gradient method applied to computing approximate minimizers of a model of the material energy for a shape memorsy material in the martensitic phase. This energy functional is not convex so special considerations are needed to assure that the method does not get stuck in the many local minima. The second method uses a multigrid based approach, averaging the approximate minimizers to include a probability distribution of the various orientations within the solution, i.e., it uses a Young measure to compute the minimizer.
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The purpose of this paper is to discuss some numerical difficulties in a two-dimensional mathematical model for displacive phase transformations. The usual approach to discretizing this model uses finite elements. Some previous work has computed certain microstructures by using a uniform square mesh discretization together with a form of derivative averaging. The overall scheme is equivalent to using bilinear elements with a one point quadrature approximation to discretize the energy functional E below. The resulting functional will be denoted by E_{h}. Although the square mesh respects a key symmetry property of the energy functional, when using this technique we have found a computational 'skewing' problem. This problem has a deleterious effect on the numerical solutions since it generates a grid scale spurious oscillation. In this paper, we will study this skewing phenomenon and attempt to show its effect on the computed microstructure.
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Magnetostriction is the phenomena of the deformation of ferromagnetic materials in external magnetic fields.3'14 In the early 1970's certain rare earth intermetallic compounds eg, TbDyFe2 (Terfenol), were discovered to exhibit high magnetostriction at room temperatures. Since then such materials have been of intense interest. There is however not yet a generally accepted coherent theoretical model and the principal mechanisms of magnetostriction still remain to be clarified. Recent advances in the understanding of microstructural behavior in modern materials1'2'6'8 have enabled James and Kinderlehrer to propose a phenomenological energy functional:10 E(y,m) = j{W(Vy(x),m(y(x))) — He m(y(x))} dx + J IVU(i)I2di (1) where y is the deformation of the material y:—1R3, w=y(11), m : w —4 1R3, and U is related to m by Maxwell's equation: V•(—VU+m,)=O inlEt3
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One and two dimensional real and complex fast fourier transform (FFT) algorithms were applied to the results of displacement phase and magnitude maps obtained from interferometric studies on a type 1-3 piezocomposite specimen electrically driven at a number of frequencies between 6 kHz and 1.0 MHz. Analysis was performed on experimental results for materials consisting of square cross section piezoelectric rods embedded in a square array within an epoxy matrix. Because the spatial periodicities of plate modes at 6 kHz frequencies were much longer than the spacing between rods, two dimensional high pass, low pass, band pass, and notch filters were used to separate plate and standing wave thickness modes. When this analysis was done on measurements taken at 353 kHz, with a only single rod active, is was possible to isolate modes associated with thickness mode vibration of the rods, vibration of the matrix rod interface, and travelling modes radiating from the rods.
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Thin plate and thin shell problems are generally set on plane reference domains with a curved boundary. Their approximation by conforming finite element methods requires C^{1}- curved finite elements entirely compatible with the associated C^{1}-rectilinear finite elements. In this contribution, we introduce a C^{1}-curved finite element compatible with the P_{5}-Argyris element, we study its approximation properties, and then, we use such an element to approximate the solution of thin plate or thin shell problems set on a plane curved boundary domain. Finally, we discuss the use of such C^{1}-curved elements to approximate junctions between thin shells.
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An electromechanical surface damping (EMSD) technique is proposed. The technique is a combination of the constrained layer damping and the shunted piezoelectric methods, where the viscoelastic layer attached to the surface of the vibrating substructure is constrained by a shunted piezoelectric ceramic element. A mathematical model of the dynamic behavior of the coupled piezoelectric/constrained layer/substructure (EMSD element) is developed, implemented into a finite element algorithm, and used to investigate the effect of some of the system parameters on the dynamic characteristics (the first three natural frequencies and modal loss factors) of a generic cantilever beam. The effect of the following system parameters is considered: storage modulus ratios, material loss factors, thickness ratios, and the axial location of the EMSD element. The algorithm is also used to demonstrate the effectiveness of the proposed EMSD technique in controlling the peak vibration amplitudes at the first two natural frequencies of the cantilever beam.
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Shape Memory Alloys (SMA's) are intermetallic materials (chemical compounds of two or more elements) that are able to sustain a residual deformation after the application of a large stress, but they 'remember' the original shape to which they creep back, without the application of any external force, after they are heated above a certain critical temperature. We present here a general one-dimensional dynamic mathematical model which reflects the balance laws for linear momentum and energy. The system accounts for thermal coupling, time-dependent distributed and boundary inputs and internal variables. Well-posedness is obtained using an abstract formulation in an appropriate Hilbert space and explicit decay rates for the associated linear semigroup are derived. Numerical experiments using finite- dimensional approximations are performed for the case in which the thermodynamic potential is given in the Landau-Devonshire form. The sensitivity of the solutions with respect to the model parameters is studied.
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A fully coupled mathematical model describing the dynamics of a cylindrical structural acoustics problem is presented. The geometry of interest consists of an acoustic field lying inside a vibrating thin cylindrical shell. In this model, the shell dynamics are coupled to the interior acoustic field through pressure and momentum conditions. Because the model will ultimately be used in control applications involving piezoceramic actuators, the loads and material contributions resulting from bonded piezoceramic patches are also included in the discussion. Strong and weak forms of the modeling set of coupled partial differential equations (PDE's) are presented, thus yielding a framework which is amenable to the application of various approximation techniques to the problem of developing schemes for forward simulations, parameter estimation, and application of PDE-based control strategies.
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In classical mathematical problems describing magnetostrictive ferromagnetic materials, one seeks a magnetization field m and deformation field y that minimize an energy functional composed of four terms: the field energy, the anisotropy or stored energy, the interaction energy, and the exchange energy. The total energy is minimized subject to boundary conditions on the deformation and a pointwise constraint on the magnitude of the deformation: m equals 1 everywhere in the body. The most important mathematical characteristic of this problem is nonattainment: minimizing sequences may oscillate or chatter, and a classical minimizer of the energy may not exist. This paper deals with a particular class of such problems based on the model for Tb_{x}Dy_{(1-x})Fe_{2} developed by James and Kinderlehrer. We seek minimizers the energy of a spherical body subjected to an applied field h_{0} that is uniform in space and a dead load generated by a constant tensor T_{0}. Exchange energy is set to zero.
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The primary factor governing response time in SMA actuators is heat transfer rate which is represented by a nonlinear differential equation for cooling by free convection. To use classical control theory for controlling the temperature of the SMA wire, the nonlinear model must be linearized. In this paper, the heat transfer rate model is linearized using Chebyshev polynomials. This method of linearization has the same desirable local stability properties as the more traditional Taylor series expansion. It is shown that the maximum errors between the actual and linearized responses and their derivatives using Chebyshev linearization are smaller in magnitude than the errors obtained using Taylor series expansion, making Chebyshev linearization more desirable for this application.
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Interfacial slippage of a piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed. A close form solution is formulated for a circular inclusion subjected to a far field in-plane electric loading and a far field out-of-plane mechanical loading. The resulting mixed boundary value problem leads to a system of dual series equations, which is then reduced to a Fredholm integral equation of the first kind. The extent of slip region along the matrix/inclusion interface, which depends on both mechanical and electrical fields, along with the distribution of shear tractions at the interface can then be determined by solving the integral equations numerically.
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A parallel processor that is optimized for real-time linear control has been developed. This modular system consists of A/D modules, D/A modules, and floating-point processor modules. The scalable processor uses up to 1,000 Motorola DSP96002 floating-point processors for a peak computational rate of 60 GFLOPS. Sampling rates up to 625 kHz are supported by this analog-in to analog-out controller. The high processing rate and parallel architecture make this processor suitable for computing state-space equations and other multiply/accumulate-intensive digital filters. Processor features include 14-bit conversion devices, low input-to-output latency, 240 Mbyte/s synchronous backplane bus, low-skew clock distribution circuit, VME connection to host computer, parallelizing code generator, and look- up-tables for actuator linearization. This processor was designed primarily for experiments in structural control. The A/D modules sample sensors mounted on the structure and the floating- point processor modules compute the outputs using the programmed control equations. The outputs are sent through the D/A module to the power amps used to drive the structure's actuators. The host computer is a Sun workstation. An OpenWindows-based control panel is provided to facilitate data transfer to and from the processor, as well as to control the operating mode of the processor. A diagnostic mode is provided to allow stimulation of the structure and acquisition of the structural response via sensor inputs.
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A performance study of a prototype programmable structure is presented. The programmable structure considered here consists of a self-sensing actuator imbedded in a beam with a self- contained surface mounted control module. The structure is powered by a 9 volt battery (external). Active control is implemented through the use of a positive position feedback filter which suppresses vibration in the beam. The results of the experimental implementation of a programmable structure are presented. An identification of the physical parameters using modal techniques is performed. These physical parameters are then used to perform a parameter study to illustrate the performance capabilities of a programmable structure. The programmable beam presented here is capable of reducing the settling time of the beam by an order of magnitude. This corresponds to an order of magnitude increase in the damping ratio. Similar results can be obtained by using passive damping treatment in the form of constrained layers. In contrast, however, the programmable structure is capable of adapting to changing operating conditions whereas passive damping treatments are designed for a single performance objective.
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Using an Artificial Neural Network (ANN) trained with the Least Mean Square (LMS) algorithm we have designed a robust linear quadratic regulator for a range of plant uncertainty. Since there is a trade-off between performance and robustness in the conventional design techniques, we propose a design technique to provide the best mix of robustness and performance. Our approach is to provide different control strategies for different levels of uncertainty. We describe how to measure these uncertainties. We will compare our multiple strategies results with those of conventional techniques e.g. H_{(infinity} ) control theory. A Lyapunov equation is used to define stability in all cases.
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Low Authority Threshold Control (LATC) is an active control strategy resulting in a piecewise continuous, with respect to time, constant gain control law which may be used for the vibration control of flexible structures. The LATC optimal control law gains are dependent on the state vector at the time the gains are applied and are defined by a two-point boundary value problem and a set of integral equality constraints. Because an iterative solution technique is required to determine the optimal gains, the real-time implementation of this control law presented certain difficulties. In this work, a neural network system is trained to determine the optimal gains in real-time for each of two experiments: a cantilevered beam and a nonlinear Duffing oscillator. The optimal gains generated by the neural network system are utilized in an LATC rate feedback control law for the vibration control of these systems.
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This paper discusses a control algorithm for active vibration suppression in flexible structures with distributed actuation. This control algorithm employs an integral transformation of the distributed-parameter system such that the transformed system behaves according to a desired second-order partial differential equation in space and time. Furthermore, a neural network is used to learn and perform this integral transformation which can be very difficult to calculate analytically and which has a kernel that is generally nonlinear. Once the system has been transformed, sliding mode control algorithms may be used to yield a closed-loop system which is insensitive to disturbances or parameter variations. An Euler-Bernoulli beam with piezoelectric actuation will be considered as an illustrative example of this particular strategy.
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A control methodology integrating sliding mode control, distributed parameter systems theory, and fuzzy control is presented for the vibration damping of flexible structures. The method reduces the theoretically infinite-order system to a second-order representation, still capturing its vibratory nature in a series of decentralized input/output loops. A sliding controller is then designed with fuzzy control gain weighting to increase the performance of the system off the sliding surface. The results of computer simulations are presented including comparisons demonstrating superior damping performance to output velocity feedback.
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Composite and porous materials often appear in nature. Many composites may be considered orthotropic such as wood or bone. The elastic behavior of these composites under shear stresses is characterized by three independent shear moduli. We consider the totality of orthotropic composites made from two isotropic linearly elastic components in fixed proportion. For a prescribed triple of shear stresses we find optimal bounds on the strongest and weakest orthotropic composites. Mathematically this problem is one of constrained optimization. The set of constraints are related to the convex hull of a surface in three dimensions. For given values of the component elasticities the bounds are computed numerically.
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The shape-memory effect is the ability of a material to recover, on heating, apparently plastic deformations that it suffers below a critical temperature. These apparently plastic strains are not caused by slip or dislocation, but by deformation twinning and the formation of other coherent microstructures by the symmetry-related variants of martensite. In single crystals, these strains depend on the transformation strain and can be quite large. However, in polycrystals made up of a large number of randomly oriented grains, the various grains may not deform cooperatively. Consequently, these recoverable strains depend on the texture and may be severely reduced or even eliminated. Thus, the shape-memory behavior of polycrystals may be significantly different from that of a single crystal. We address this issue by studying some model problems in the setting of anti-plane shear.
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The effective elastic and dielectric moduli of a composite made from piezoelectric materials are examined, with particular emphasis on applications to unpoled piezoelectric ceramics and layered materials. Explicit formulae for the effective moduli and coupling of a layered material are derived. A self-consistent estimate of the moduli of an isotropic polycrystal is obtained through an effective medium approximation (EMA), which takes into account the interaction between each individual grain and the surrounding composite. This estimate shows that the grains behave as uncoupled grains with electric and elastic constants modified by the behavior of surrounding grains. A similar effect is also observed in bounds (established via classical variational principles) on the moduli of a statistically isotropic polycrystal. Numerical implementation of the EMA and bounds show good agreement with data for unpoled barium titanate ceramic. For a general composite with piezoelectric constituents, it is shown that the effective electromechanical coupling can be bounded by the largest coupling factor of the components.
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Composites of piezoelectric and magnetostrictive materials show a product property called magnetoelectricity which is absent in its constituent phases. The electric and magnetic fields are linked in the composite through the elastic stress-strain fields of the piezoelectric and magnetostrictive phases. Thus an applied magnetic field causes electric polarization or an electric field applied across the composite causes magnetization in the composite material. Such materials are potentially useful as magnetoelectric transducers. In this work, we report the results of a theoretical and experimental investigation of ME composites of PZT-CoFe_{2}O_{4} and BaTiO_{3}-CoFe_{2}O_{4} with various connectivities. The magnetically induced ME effect was measured by applying an ac magnetic field of 1 kHz frequency with a variable dc magnetic bias field. The maximum values of the ME voltage coefficient, were 92.8 (V/m)(kA/m) for the 2-2 CoFe_{2}O_{4}-PZT4 composites and 9.55 (V/m)(kA/m) for CoFe_{2}O_{4}-BaTiO_{3} (20:80 mole %) ball mill mixed composites. Theoretical models were developed to calculate the ME voltage coefficient of 2-2 composites with different boundary conditions. Composites with 3-0 and 0-3 connectivities were modeled using a cubes model. The results show that the connectivities have a great effect on the magnetoelectric properties. An improvement of more than two orders of magnitude is possible by proper selection of materials and better process control to tailor the connectivity.
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We present a new numerical method for the solution of problems of diffraction of light by a singly or doubly periodic interface between two materials. Our basic result is that the diffracted fields behave analytically with respect to variations of the interface, so that they can be represented by convergent series in powers of the height of the grating profile. A second element in the theory consists of a simple algebraic recursive formula with allows us to obtain the power series by considering a sequence of diffraction problems with flat interface. Once the Taylor coefficients have been computed, we use Pade approximants to extract the values of the fields from their power series expansions. This results in accurate predictions for the efficiencies in the resonance region; in many cases these values are several orders of magnitude more accurate than those obtained by currently available methods. For three dimensional biperiodic gratings, the performance of our method is of the same quality as far as for the singly periodic case. We demonstrate the wide applicability and accuracy of our algorithm with numerical results for two- and three-dimensional problems, and we compare our predictions with some experimental data.
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This paper concerns the problem of designing a periodic interface between two homogene0us 'materials in such a way that normally incident time-harmonic plane waves scattered from the structure have a specified far-field phase and intensity pattern. The motivation comes from applications in micro-optics where such structures are increasingly used in advanced devices. The design problem can be formulated in various ways as a mathematical optimization problem, with the Helmholtz equation as the underlying model for wave propagation. We consider a "relaxed" formulation of the design problem in which "mixtures" of the two materials are included as admissible designs. A key feature of the problem is the extreme ill-posedness: we are trying to determine a function (the refractive index) from a few scalar values (the far-field pattern). In principle there are infinitely many solutions. However, in contrast to the typical situation in inverse problems, for the design problem non-uniqueness and instability are in some sense an asset: they allow some flexibility to choose designs which are more desirable from an engineering point of view. Exploiting this flexibility presents two primary challenges: first, to characterize the source of the ill-posedness and second, to devise appropriate computational schemes. We briefly discuss both issues and present a particular computational scheme--based on the minimization of the total variation of the design-along with some numerical results.
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It is demonstrated that the planar reorientation of a free-free beam in zero gravity space can be accomplished by periodically changing the shape of the beam using embedded electromechanical actuators. The dynamics which determine the shape of the free-free beam is assumed to be characterized by the Euler-Bernoulli equation, including material damping, with appropriate boundary conditions. The coupling between the rigid body motion and the flexible motion is explained using the angular momentum expression which includes rotatory inertia and kinematically exact effects. A control scheme is proposed where the embedded actuators excite the flexible motion of the beam so that it rotates in the desired sense with respect to a fixed inertial reference. Relations are derived which relate the average rotation rate to the amplitudes and the frequencies of the periodic actuation signal and the properties of the beam. These reorientation maneuvers can be implemented by using feedback control.
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The application of shape memory alloy materials as actuators and sensors in the active control of flexible structures has been extensively reported in the literature. The design of active controllers plays an important role in the overall development of smart structures for a given application. To design active controllers for flexible structures, a mathematical representation of the system is needed. The process of constructing a model to describe the vibration properties of a structure based on experimental test data is known as structural identification method. To account for any uncertainties in the structural models and to accomplish good closed loop system performance and noise suppression properties, we have developed robust control design methodologies for flexible structures. We have utilized the eigensystem realization algorithm (ERA) for system identification and linear quadratic Gaussian with loop transfer recovery (LQG/LTR) method for designing robust controllers for a simple cantilever beam test article. The shape memory alloy, NiTiNOL, is used as an actuator. The LQG/LTR method has been modified to accommodate the limited control force provided by the actuators. The closed loop performance of the cantilever beam is experimentally determined for various types of uncertainties. The properties of robust controllers are demonstrated.
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The design and implementation of control strategies for large, flexible smart structures presents challenging problems. One of the difficulties arises in the approximation of high- order finite element models with low order models. Another difficulty in controller design arises from the presence of unmodeled dynamics and incorrect knowledge of the structural parameters. In this paper, the balance-truncation reduced-order models are employed in deriving lower-order models for complex smart structures. These methods do not introduce any spill-over problems in the closed-loop response of the system. The simplified analytical models are compared with models developed by structural identification techniques based on vibration test data. To minimize the effects of uncertainties on the closed-loop system performance of smart structures, robust control methodologies have been employed in the design of controllers. The reduced order models are employed in the design of robust controllers. To demonstrate the capabilities of shape-memory-alloy actuators, we have designed and fabricated a three-mass test article with multiple shape-memory-alloy (NiTiNOL) actuators. Generally, the non-collocation of actuators and sensors presents difficulties in the design of controllers. Controllers for a test article with non-collocated sensors and actuators are designed, implemented and tests. The closed-loop system response of the test article with two actuators and sensors has been experimentally determined and presented in the paper.
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An integrated means for active controller design and structure redesign is presented. The techniques of covariance control are used to parametrize all possible combinations of active controllers/structure redesign parameters which can stabilize the plant, and achieve certain closed-loop performance.
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An important problem in the applications is the controllability for perturbed systems. A special kind of perturbed systems, depending on a small parameter (epsilon) is the wave equation when the coefficients are rapidly oscillating or when posed in periodically perforated domains. Then a limit system (in the homogenization theory sense) can be defined. We present here a review of some recent results for exact and approximate controllability in these situations.
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Recent years have seen the successful use of continuum models--based on the minimization of an appropriate nonconvex free energy--to explain some of the complicated microstructure which is commonly observed in shape memory alloys, magnetostrictive materials, ferro- electrics, etc. However, due to their inherent static nature such models cannot provide insight into important dynamic phenomena such as: the growth of microstructure, the evolution of phase boundaries, or the eventual fate of metastable structures. We describe the behavior of some simple dynamical models of shape-memory alloys which display the formation of microstructure. The focus is on models whose underlying bulk free energy do not attain a minimum, possessing sequences of configurations with finer and finer microstructure that converge weakly to nonminimizing states. The effects of viscoelastic damping on the stability of steady state solutions, the propagation of phase boundaries, and the long-time approach to minimum energy states is discussed.
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Consider a material whose physical properties are controllable, with two possible states. Suppose further that the control can be applied in a distributed manner, e.g. by laying down a very fine grid and specifying the state of each cell independently. Can we design 'smart structures' by adjusting the control to optimize some performance criterion? This problem is difficult because the set of possible controls is discrete. There is a natural way to make it continuous, however, known as the 'relaxed approach.' Physically, it amounts to the introduction of composite materials as structural components. Mathematically, it amounts to the introduction of fluttering controls. It has evolved over the past 15 years, through the combined effort of many individuals. This is a review paper. The goal is not to present new mathematical results, but rather to publicize the technique of relaxation. Perhaps some readers will known of new problems where this technique could be used to advantage.
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A theory for displacive phase transitions where the state in the sense of phase is characterized by an order parameter and phase interfaces are identified with thin transition zones across which the strain and order parameter exhibit large gradients is presented. Connections between this theory and more standard continuum approaches that identify phase interfaces with surfaces of strain discontinuity are discussed.
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In this work we discuss two free boundary problems which arise in semiconductor processing. The first problem is concerned with deposition of titanium silicide over a wafer, a process which results in three layers of chemicals: Ti, TiSi and TiSi_{2} and three free-boundary interfaces. The second problem is concerned with the growth of loop dislocations in crystal, which develop as a result of implanting impurities and rapid thermal annealing. Existence results and properties of the solutions are discussed.
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The smartness of a shape-memory material is a consequence of its ability to form a flexible variant structure at one temperature while recognizing only a homogeneous equilibrium at a different temperature. The fine scale morphology or microstructure of this variant structure has a clear role in the macroscopic behavior of the material. To investigate these phenomena, two issues are paramount. First, the presence of several stable variants at a given temperature reflects a complicated potential well structure for the free energy of the material. Second, the presence of spatially oscillatory behavior at the small scale suggests competition between the free energy of the material and loading or other environmental effects. Both of these features represent highly nonlinear processes and thus it is to nonlinear analysis we turn for methods to successfully describe these systems. In this report we describe in an expository fashion one such technique which has been applied in several instances especially related to certain alloys or other crystalline materials.
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