Reference Domain Research Articles

Analysis and assessment of a monolithic FSI finite element method

In this paper we analyze and investigate numerically a monolithic finite element method for the incompressible Navier–Stokes equations coupled to the Saint Venant–Kirchhoff hyperelastic model. The approach strongly enforces the coupling conditions on the fluid–structure interface and treats both solid and fluid equations in a reference domain accounting for geometric motion through time-dependent coefficients. The paper derives an energy balance for the fully discrete system and proves that the finite element method is numerically stable. The performance of the method is further assessed by validating numerical results for the pressure impulse prorogation test and against the results of a recently proposed experimental 3D fluid-structure interaction benchmark problem.

Maximal regularity of the Stokes operator in an exterior domain with moving boundary and application to the Navier–Stokes equations

Consider the (Navier–) Stokes system on an exterior domain with moving boundary and Dirichlet boundary conditions. In 2003 Saal proved that the Stokes operator in a domain with moving boundary has the property of maximal regularity provided that the operator is invertible. Hence his result can be applied if the domain is bounded or by adding a shift to the Stokes operator if the domain is unbounded or the time interval is finite. In this paper, we will generalize his result to a result global in time if the reference domain is an exterior domain. Finally, we will apply this result to the Navier–Stokes equations to obtain a global in time existence theorem for small data.

Rapid deployment of curved surfaces via programmable auxetics

Deployable structures are physical mechanisms that can easily transition between two or more geometric configurations; such structures enable industrial, scientific, and consumer applications at a wide variety of scales. This paper develops novel deployable structures that can approximate a large class of doubly-curved surfaces and are easily actuated from a flat initial state via inflation or gravitational loading. The structures are based on two-dimensional rigid mechanical linkages that implicitly encode the curvature of the target shape via a user-programmable pattern that permits locally isotropic scaling under load. We explicitly characterize the shapes that can be realized by such structures---in particular, we show that they can approximate target surfaces of positive mean curvature and bounded scale distortion relative to a given reference domain. Based on this observation, we develop efficient computational design algorithms for approximating a given input geometry. The resulting designs can be rapidly manufactured via digital fabrication technologies such as laser cutting, CNC milling, or 3D printing. We validate our approach through a series of physical prototypes and present several application case studies, ranging from surgical implants to large-scale deployable architecture.

Vibration of FG-GPLs eccentric annular plates embedded in piezoelectric layers using a transformed differential quadrature method

In this paper, a transformed differential quadrature method (TDQM) for the free vibration analysis of functionally graded (FG) multilayer nanocomposite eccentric annular plates reinforced by graphene nanoplatelets (GPLs) and embedded in piezoelectric layers is developed. In this regards, a suitable conformal mapping is utilized to calculate the transformed weighting coefficients of the differential quadrature method (DQM) in physical domain which enables one to directly discretize the partial differential equations in the reference domain. The modified Halpin–Tsai model is employed to estimate the effective mechanical properties of the nanocomposites. The governing equations are derived based on the first-order shear deformation theory (FSDT) using Hamilton’s principle and Maxwell’s equation. The numerical results show that the TDQM has a very good convergence rate. Furthermore, the accuracy and reliability of the proposed method are verified by comparing the results with those obtained by simulating some problems via ABAQUS software and also, in the limit cases with those available in the literature. The effects of GPLs weight fractions, number of GPLs reinforced layers, distribution types of GPLs, different boundary conditions and various plate geometrical parameters such as the plate thickness, piezoelectric layers thickness, external electrical voltage and off-set parameter on the natural frequency parameters are investigated.

Designing polymer spin packs by tailored shape optimization techniques

We consider optimal shape design problems for polymer spin packs which are widely used in the production of synthetic fibers and nonwoven materials. The design goal is the minimization of the residence time of the polymer, which can be achieved by adjusting the wall shear stress along the boundary. Depending on the specific industrial setting we construct two tailored algorithms. First, we consider the design in three spatial dimensions based on a PDE constrained shape optimization problem. Here, the constraint is given by the Stokes flow. Second, we change the design goal and want to construct shapes in two spatial dimensions which allow for a lower bound on the wall shear stress. This can be incorporated as an additional state constraint. By relaxing this condition and employing the method of mapping we can pull-back the problem onto a fixed reference domain. We get an elegant formulation of this state constrained optimization problem, in which geometric constraints on the boundary can also be included. After discretization we end up with a large-scale NLP which can be handled by existing solvers. Finally, we present numerical results underlining the feasibility of our approach.

Segmentation of parotid glands from registered CT and MR images

PurposeTo develop an automatic multimodal method for segmentation of parotid glands (PGs) from pre-registered computed tomography (CT) and magnetic resonance (MR) images and compare its results to the results of an existing state-of-the-art algorithm that segments PGs from CT images only. MethodsMagnetic resonance images of head and neck were registered to the accompanying CT images using two different state-of-the-art registration procedures. The reference domains of registered image pairs were divided on the complementary PG regions and backgrounds according to the manual delineation of PGs on CT images, provided by a physician. Patches of intensity values from both image modalities, centered around randomly sampled voxels from the reference domain, served as positive or negative samples in the training of the convolutional neural network (CNN) classifier. The trained CNN accepted a previously unseen (registered) image pair and classified its voxels according to the resemblance of its patches to the patches used for training. The final segmentation was refined using a graph-cut algorithm, followed by the dilate-erode operations. ResultsUsing the same image dataset, segmentation of PGs was performed using the proposed multimodal algorithm and an existing monomodal algorithm, which segments PGs from CT images only. The mean value of the achieved Dice overlapping coefficient for the proposed algorithm was 78.8%, while the corresponding mean value for the monomodal algorithm was 76.5%. ConclusionsAutomatic PG segmentation on the planning CT image can be augmented with the MR image modality, leading to an improved RT planning of head and neck cancer.

Variations of Deformations with Self-Contact on Lipschitz Domains

We analyze the injective deformations on Lipschitz domains, motivated by the configurations of solid elastic materials. Our goal is to rigorously derive the first order optimality conditions when minimizing over deformations with a self-contact constraint. In a previous work, we established a variational equation for minimizers of a second-gradient hyper-elastic energy. This involved obtaining a Lagrange multiplier for the self-contact constraint as a nonnegative surface measure multiplied by the outward normal vector. For that approach, we had to assume that the boundary of the reference domain is smooth. In this work, we carry out an analogous characterization assuming only that the boundary is locally the graph of a Lipschitz function. These Lipschitz domains possess everywhere defined interior cones, which we use to characterize the displacements that produce paths of injective deformations. Our application to nonlinear elasticity results in a vector-valued surface measure that is characterized by the local nonsmooth geometry of the domain at the points of self-contact.

Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers

We propose a solution strategy for a multimaterial minimum compliance topology optimization problem, which consists in finding the optimal allocation of a finite number of candidate (possibly anisotropic) materials inside a reference domain, with the aim of maximizing the stiffness of the body. As a relevant and novel application we consider the optimization of self-assembled structures obtained by means of diblock copolymers. Such polymers are a class of self-assembling materials which spontaneously synthesize periodic microstructures at the nanoscale, whose anisotropic features can be exploited to build structures with optimal elastic response, resembling biological tissues exhibiting microstructures, such as bones and wood. For this purpose we present a new generalization of the classical Optimality Criteria algorithm to encompass a wider class of problems, where multiple candidate materials are considered, the orientation of the anisotropic materials is optimized, and the elastic properties of the materials are assumed to depend on a scalar parameter, which is optimized simultaneously to the material allocation and orientation. Well-posedness of the optimization problem and well-definition of the presented algorithm are narrowly treated and proved. The capabilities of the proposed method are assessed through several numerical tests.

Computing parametrized solutions for plasmonic nanogap structures

The interaction of electromagnetic waves with metallic nanostructures generates resonant oscillations of the conduction-band electrons at the metal surface. These resonances can lead to large enhancements of the incident field and to the confinement of light to small regions, typically several orders of magnitude smaller than the incident wavelength. The accurate prediction of these resonances entails several challenges. Small geometric variations in the plasmonic structure may lead to large variations in the electromagnetic field responses. Furthermore, the material parameters that characterize the optical behavior of metals at the nanoscale need to be determined experimentally and are consequently subject to measurement errors. It then becomes essential that any predictive tool for the simulation and design of plasmonic structures accounts for fabrication tolerances and measurement uncertainties.In this paper, we develop a reduced order modeling framework that is capable of real-time accurate electromagnetic responses of plasmonic nanogap structures for a wide range of geometry and material parameters. The main ingredients of the proposed method are: (i) the hybridizable discontinuous Galerkin method to numerically solve the equations governing electromagnetic wave propagation in dielectric and metallic media, (ii) a reference domain formulation of the time-harmonic Maxwell's equations to account for arbitrary geometry variations; and (iii) proper orthogonal decomposition and empirical interpolation techniques to construct an efficient reduced model. To demonstrate effectiveness of the models developed, we analyze geometry sensitivities and explore optimal designs of a 3D periodic coaxial nanogap structure.

Reference Abstract Domains and Applications to String Analysis

Abstract interpretation is a well established theory that supports reasoning about the run-time behaviour of programs. It achieves tractable reasoning by considering abstractions of run-time states, rather than the states themselves. The chosen set of abstractions is referred to as the abstract domain. We develop a novel framework for combining (a possibly large number of) abstract domains. It achieves the effect of the so-called reduced product without requiring a quadratic number of functions to translate information among abstract domains. A central notion is a reference domain, a medium for information exchange. Our approach suggests a novel and simpler way to manage the integration of large numbers of abstract domains. We instantiate our framework in the context of string analysis. Browser-embedded dynamic programming languages such as JavaScript and PHP encourage the use of strings as a universal data type for both code and data values. The ensuing vulnerabilities have made string analysis a focus of much recent research. String analysis tends to combine many elementary string abstract domains, each designed to capture a specific aspect of strings. For this instance the set of regular languages, while too expensive to use directly for analysis, provides an attractive reference domain, enabling the efficient simulation of reduced products of multiple string abstract domains.

Modeling shape selection of buckled dielectric elastomers

A dielectric elastomer whose edges are held fixed will buckle, given a sufficiently applied voltage, resulting in a nontrivial out-of-plane deformation. We study this situation numerically using a nonlinear elastic model which decouples two of the principal electrostatic stresses acting on an elastomer: normal pressure due to the mutual attraction of oppositely charged electrodes and tangential shear (“fringing”) due to repulsion of like charges at the electrode edges. These enter via physically simplified boundary conditions that are applied in a fixed reference domain using a nondimensional approach. The method is valid for small to moderate strains and is straightforward to implement in a generic nonlinear elasticity code. We validate the model by directly comparing the simulated equilibrium shapes with the experiment. For circular electrodes which buckle axisymetrically, the shape of the deflection profile is captured. Annular electrodes of different widths produce azimuthal ripples with wavelengths that match our simulations. In this case, it is essential to compute multiple equilibria because the first model solution obtained by the nonlinear solver (Newton's method) is often not the energetically favored state. We address this using a numerical technique known as “deflation.” Finally, we observe the large number of different solutions that may be obtained for the case of a long rectangular strip.

Linguistic terms with weakened hedges: A model for qualitative decision making under uncertainty

When expressing the experts’ opinions in qualitative decision making (QDM), linguistic hedges can be considered to modify the force expressed by a predefined linguistic term. If an expert is not sure to select one term, weakened hedges would be a natural way to express the uncertainty. This is usually implemented by using a hedge to modify the most possible term, like the expression “more or less good”. To model the uncertainty implied by hedges in QDM, this paper presents a novel linguistic representational and computational model in which the linguistic expressions take the form of a weakened hedge and a linguistic term, which is named as linguistic term with weakened hedge (LTWH). The syntax of LTWHs is defined by a set of hedges and a set of linguistic terms. The semantics of a LTWH is determined, objectively, based on the semantics of the term and a similarity measure of the reference domain. Accordingly, the negation, order relations and some basic operations of LTWHs are defined. To illustrate the effectiveness of LTWHs in granular computing, the connection to some multi-granularity linguistic models is exploited and a process for unifying multi-granularity linguistic information is developed. The major contritions of this paper are: (1) The proposed model enables a new manner to express and operate uncertain linguistic information in QDM; (2) it possesses clear syntax and semantics and the computational results are very interpretable; and (3) the proposed solution of multi-granularity linguistic unification maintains the semantics of the original linguistic information.

A non-standard finite element method for dynamical behavior of cylindrical classical shell model

This paper deals with the approximation of the dynamical behaviors of a cylindrical shell, modeled by the classical Naghdi shell’s mathematical model over a reference domain. We use a non-standard method so called MITC4-p finite element method, which comes from non standard mixed interpolated tensorial component (MITC) formulation for Reissner–Mindlin plates. The performance of this approach is assessed for both, the modal analysis of the free vibrations modes, and time-domain analysis of vibration responses. Some numerical experiments are presented to show the performance of the method, in particular, it is numerically observed that the method is completely locking free and independent of the thickness.

Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity

The size-dependent longitudinal and torsional dynamic problems for small-scaled rods are modeled by utilizing an integral formula of two-phase nonlocal theory. The energies diffused from surrounding particles in a reference domain can be taken into account in the rod model by using the two-phase nonlocal theory, which depends on the internal characteristic length via convolution integrals over exponential kernel. Unlike the nonlocal differential models, which are inconsistent as for some rod- and beam-type problems, the developed nonlocal integral models are both self-consistent and well-posed. The governing equations and boundary conditions for the longitudinal and torsional dynamics of the nonlocal rods are deduced by employing the Hamilton principle. By reducing the complicated integro-differential equations to a fourth order differential equation with mixed boundary conditions, the asymptotic solutions of predicting the longitudinal and torsional frequencies are derived for the two-phase nonlocal rod under clamped-clamped and clamped-free boundary conditions. The closed-form solutions for longitudinal and torsional dispersion relations are obtained. The single-walled carbon nanotube, single layer graphene sheet and silicon are chosen as nanoscaled rods to study the size-dependent effect on the dispersion relation and vibration frequencies, which can show a good agreement with molecular dynamics results or experimental data.

On longitudinal dynamics of nanorods

The longitudinal dynamic problem of a size-dependent elasticity rod is formulated by utilizing an integral form of nonlocal strain gradient theory. The nonlocal strain gradient model accounts for the energies diffused from surrounding particles in a reference domain by utilizing the convolution integral over nonlocal kernel functions, and can account for micro/nano-structures with internal displacement field via gradient forms. Unlike the size-dependent differential models, the developed integral model is both self-consistent and well-posed. The governing equations and boundary conditions for the longitudinal dynamics of the rod are deduced by employing the Hamilton principle. In addition to the well-known classical boundary conditions, the developed integral rod model also contains non-classical boundary conditions. By reducing the complicated integro-differential equations to a sixth order differential equation with mixed boundary conditions, the asymptotic solutions for predicting the natural frequencies of the rods are derived for the nonlocal strain gradient rod under various boundary conditions. It is shown explicitly that the integral rod model can exert stiffness-softening and stiffness-hardening effects by considering various values of the size-dependent parameters. By studying the size-dependent effects on the longitudinal dynamics of monolayer graphene, the dispersion relation calculated by using the nonlocal strain gradient model can show good agreement with the experimental data obtained by inelastic X-ray scattering. The size-dependent effect can make monolayer graphenes possess softening frequencies.

Case report of infrequent variation of third head of biceps brachii muscle

The numerical analysis of laminated composite plates and shells resting on nonlinear elastic foundation is the main topic of the paper. The generalized differential quadrature (GDQ) technique and the Newton-Raphson iteration are employed to obtain the solution of the static problems under consideration. The nonlinear elastic foundation is modeled using the Winkler-Pasternak model embedded with quadratic and cubic nonlinearities in order to a have a more complete description of the interaction. The structural behavior is modeled by means of higher-order displacement fields developed in the framework of a unified formulation. Several lamination schemes are studied. The class of sandwich structures with an inner soft-core is also taken into account with the help of the Murakami’s function, which correctly captures the so-called zig-zag effect. The presented approach can deal with doubly-curved surfaces characterized by two radii of curvature that can vary in each point of the reference domain, whereas most of the examples available in the literature considers only shells with constant curvature, such as spherical and cylindrical shells. Solutions are presented in terms of through-the-thickness variations of strains, stresses, and displacements. For these purposes, a posteriori recovery procedure based on the GDQ method is introduced. The accuracy and effectiveness of the proposed approach are proven by means of the comparison with the numerical results obtained by a three-dimensional finite element model.

Linear Static Behavior of Damaged Laminated Composite Plates and Shells.

A mathematical scheme is proposed here to model a damaged mechanical configuration for laminated and sandwich structures. In particular, two kinds of functions defined in the reference domain of plates and shells are introduced to weaken their mechanical properties in terms of engineering constants: a two-dimensional Gaussian function and an ellipse shaped function. By varying the geometric parameters of these distributions, several damaged configurations are analyzed and investigated through a set of parametric studies. The effect of a progressive damage is studied in terms of displacement profiles and through-the-thickness variations of stress, strain, and displacement components. To this end, a posteriori recovery procedure based on the three-dimensional equilibrium equations for shell structures in orthogonal curvilinear coordinates is introduced. The theoretical framework for the two-dimensional shell model is based on a unified formulation able to study and compare several Higher-order Shear Deformation Theories (HSDTs), including Murakami’s function for the so-called zig-zag effect. Thus, various higher-order models are used and compared also to investigate the differences which can arise from the choice of the order of the kinematic expansion. Their ability to deal with several damaged configurations is analyzed as well. The paper can be placed also in the field of numerical analysis, since the solution to the static problem at issue is achieved by means of the Generalized Differential Quadrature (GDQ) method, whose accuracy and stability are proven by a set of convergence analyses and by the comparison with the results obtained through a commercial finite element software.

Cut finite element methods for elliptic problems on multipatch parametric surfaces

We develop a finite element method for the Laplace–Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a subdomain of the unit square which is bounded by a number of smooth trim curves. A patchwise tensor product mesh is constructed by using a structured mesh in the reference domain. Since the patches are trimmed we obtain cut elements in the vicinity of the interfaces. We discretize the Laplace–Beltrami operator using a cut finite element method that utilizes Nitsche’s method to enforce continuity at the interfaces and a consistent stabilization term to handle the cut elements. Several quantities in the method are conveniently computed in the reference domain where the mappings impose a Riemannian metric. We derive a priori estimates in the energy and L2 norm and also present several numerical examples confirming our theoretical results.

Opinions Regarding the Fundamental Domains of National Security

Abstract “The multiple dimension of national security in the international contemporary context - characterized by accelerated, radical changes - is determined by the necessity of constantly promoting and permanently defending our national values and interests against the extraordinary diversity of insecurity factors - destabilizing vectors that manifest themselves in a surprising manner and create difficulties in the process of assessment of the specific disruptive effects. This complex framework of social evolution offers numerous perspectives for the analysis of the concept of security, but these studies do not need to dilute its fundamental connotations, but it is necessary for these efforts to particularly relate to the subsystems that are really relevant for individual and collective security and maintenance and, on the basis of a very good documentation, be exactly circumscribed into the legislative coordinates that correspond to the reference domain.”

A posteriori stress and strain recovery procedure for the static analysis of laminated shells resting on nonlinear elastic foundation

Abstract The numerical analysis of laminated composite plates and shells resting on nonlinear elastic foundation is the main topic of the paper. The generalized differential quadrature (GDQ) technique and the Newton-Raphson iteration are employed to obtain the solution of the static problems under consideration. The nonlinear elastic foundation is modeled using the Winkler-Pasternak model embedded with quadratic and cubic nonlinearities in order to a have a more complete description of the interaction. The structural behavior is modeled by means of higher-order displacement fields developed in the framework of a unified formulation. Several lamination schemes are studied. The class of sandwich structures with an inner soft-core is also taken into account with the help of the Murakami’s function, which correctly captures the so-called zig-zag effect. The presented approach can deal with doubly-curved surfaces characterized by two radii of curvature that can vary in each point of the reference domain, whereas most of the examples available in the literature considers only shells with constant curvature, such as spherical and cylindrical shells. Solutions are presented in terms of through-the-thickness variations of strains, stresses, and displacements. For these purposes, a posteriori recovery procedure based on the GDQ method is introduced. The accuracy and effectiveness of the proposed approach are proven by means of the comparison with the numerical results obtained by a three-dimensional finite element model.