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Norm-resolvent convergence with an order-sharp error estimate is established for Neumann Laplacians on thin domains in Rd, d≥2, converging to metric graphs in the limit of vanishing thickness parameter in the “resonant” case. The vertex matching conditions of the limiting quantum graph are revealed as being closely related to those of the δ′ type.

In this study, structural thin-layer sandwich walls (SWs) made of steel-fibre-reinforced concrete (SFRC) without conventional reinforcements were investigated. Other researchers have shown that SWs with thin wythes can be used as load bearing structures in low-rise buildings, thereby reducing the amount of concrete by 2–5 times if compared to conventional reinforced-concrete SWs. In most studies, relatively warm climatic regions are the focus, and thin-layer SWs with shear connectors to obtain a certain level of composite action are investigated. In almost no studies has sound insulation been evaluated. In this study, a numerical investigation of structural, thermal and sound insulation performances was carried out. The load-bearing capacities of composite and non-composite SWs are compared. Regions with the lowest five-day mean air temperature of −20 ∘ C were considered. The characteristics of the SW are compared to the requirements given in relevant European and Latvian standards. The minimum thermal insulation for family houses varies from 120 mm to 200 mm, depending on the material. To ensure sufficient sound insulation, the average thickness of the concrete wythes should be around 60 mm, preferably with a 15 mm difference between them. Structural analysis of the proposed wall panel was performed using non-linear finite element analysis software ATENA Science. The obtained load-bearing capacity exceeded the design loads of a single-story family house by around 100 times, regardless of the degree of composite action. [ABSTRACT FROM AUTHOR]

Due to the advancement in the development of semiconductors used in the power converters, the printed circuit boards (PCBs) require an in-depth study of their electromagnetic behavior. To characterize the behavior of the PCBs, the Darwin model is employed, which can take into account all the coupled effects, namely resistive, inductive, and capacitive effects, at the intermediate frequencies. Nevertheless, the study of particular structures having a geometric dimension smaller than the others can create meshing difficulties. The modeling of thin structures by the finite element method requires the optimization of the mesh. To circumvent this issue, the shell elements for both node and edge elements are applied in this work. Finally, to validate the proposed approaches, two PCBs with different geometries are studied in both time and frequency domains, where the measurements for a single PCB are provided to compare with the numerical results. [ABSTRACT FROM AUTHOR]

We consider a class of models for nonlinearly elastic surfaces in this work. We have in mind thin, highly deformable structures modeled directly as two-dimensional nonlinearly elastic continua, accounting for finite membrane and bending strains and thickness change. We assume that the stored-energy density is polyconvex with respect to the second gradient of the deformation, and we require that it grow unboundedly as the local area ratio approaches zero. For sufficiently fast growth, we show that the latter is uniformly bounded away from zero at an energy minimizer. With this in hand, we rigorously derive the weak form of the Euler-Lagrange equilibrium equations. [ABSTRACT FROM AUTHOR]

Among the metal matrix composite (MMC) materials, aluminum carbon fiber reinforced (Al-CF) composites are suitable for thermal management applications in fields, such as space, automobiles, etc. However, the application of Al-CF composite material is limited due to manufacturing difficulties, such as brittleness, non-uniform dispersion of fibers, and failure under low loads being a soft material. Challenges further increase in the case of producing thin-walled structures and features with a high aspect ratio (e.g. 2D structures).In this study, an experimental investigation is carried out on the machinability of Al-CF composite material. The material response to AWJs is assessed by studying the kerf characteristics in through-cutting and analyzing the machined surface micrographs. From the results, it is observed that the kerf width and the erosion depth increase with the waterjet pressure (P) and abrasive mass flow rate (mf), and decreased with the jet traverse rate (vf). A minimum taper is obtained at low mf and low vf at an optimum P. The material removal rate is maximum at high P and mf at an optimum vf. Based on this understanding, fabrication of a thin-wall (2D) structure with small features in Al-CF composite material is demonstrated by micro-AWJs. [ABSTRACT FROM AUTHOR]

This paper focuses on the weakly singular integrals in thin structures of 3D boundary element method. In the pioneering woks for the weakly singular integrals, the singularities are eliminated through introducing transformations whose Jacobian at the singular point is zero. When using these methods for weakly singular boundary integrals on elements with a large-angle and large side length ratio, near singularity will appear. In this paper, a dual transformation method is proposed to accurately calculate weakly singular integrals in which the near singularity is also considered. The numerical implementation is divided into several steps. Firstly, the (α , β) coordinate transformation method is used to eliminate the weak singularity by introducing the α factors in the transformation Jacobian. Then extract the integral form with near singularities about β. Finally, a corresponding distance transformation is constructed to eliminate its near singularity. The results show that the proposed method can accurately integrate for arbitrary element shape and source point position. The dual transformation method may be a good choice to evaluate weakly singular integral for thin structural problem. [ABSTRACT FROM AUTHOR]

In this tutorial, we provide a coordinate-free derivation of the system of equations that govern equilibrium of a thin shell that can undergo shear. This system involves tensorial fields representing the internal force and couple per unit length that adjacent parts of the shell exchange at their common boundary. By an appropriate decomposition of those quantities, we obtain a representation of the internal power in terms of time derivatives of suitable strain measures. Subsequently, we propose constitutive equations that employ these strain measures as independent variables. After specializing the theory to the case of unshearable shells, we linearize the resulting equations. As an application, we study the free vibrations of a pressurized spherical shell, showcasing the advantages of a coordinate-free perspective, which simplifies both the deduction and the solution of the final governing equations. • Equilibrium equations for thin shells are derived via a coordinate-free approach. • A direct approach has been employed. • Mechanical insight is facilitated by the use of the direct notation. • The effectiveness of the approach is showcased by solving a free-vibration problem. [ABSTRACT FROM AUTHOR]

In this study, structural thin-layer sandwich walls (SWs) made of steel-fibre-reinforced concrete (SFRC) without conventional reinforcements were investigated. Other researchers have shown that SWs with thin wythes can be used as load bearing structures in low-rise buildings, thereby reducing the amount of concrete by 2–5 times if compared to conventional reinforced-concrete SWs. In most studies, relatively warm climatic regions are the focus, and thin-layer SWs with shear connectors to obtain a certain level of composite action are investigated. In almost no studies has sound insulation been evaluated. In this study, a numerical investigation of structural, thermal and sound insulation performances was carried out. The load-bearing capacities of composite and non-composite SWs are compared. Regions with the lowest five-day mean air temperature of −20 ∘C were considered. The characteristics of the SW are compared to the requirements given in relevant European and Latvian standards. The minimum thermal insulation for family houses varies from 120 mm to 200 mm, depending on the material. To ensure sufficient sound insulation, the average thickness of the concrete wythes should be around 60 mm, preferably with a 15 mm difference between them. Structural analysis of the proposed wall panel was performed using non-linear finite element analysis software ATENA Science. The obtained load-bearing capacity exceeded the design loads of a single-story family house by around 100 times, regardless of the degree of composite action.

We consider the steady Bingham flow in a two-dimensional thin Y-like shaped structure, with no-slip boundary conditions and under the action of given external forces. After passage to the limit with respect to a small parameter related to the thickness of the domain, we obtain three uncoupled problems. Each of these problems describes an anisotropic flow, corresponding to a lower-dimensional “Bingham-like” constitutive law. These results are in accordance with the engineering models. [ABSTRACT FROM AUTHOR]

In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a 1 D -dimensional energy for elastic strings by using techniques from Γ-convergence. [ABSTRACT FROM AUTHOR]

Abstract The time periodic Navier–Stokes equations are considered in the three-dimensional and two-dimensional settings with Dirichlet boundary conditions in thin tube structures. These structures are finite union of thin cylinders (thin rectangles in the case of dimension two), where the small parameter ε is the ratio of the hight and the diameter of the cylinders. We consider the case of finite or big coefficient before the time derivative. This setting is motivated by hemodynamical applications. Theorems of existence and uniqueness of a solution are proved. Complete asymptotic expansion of a solution is constructed and justified by estimates of the difference of the exact solution and truncated series of the expansion in norms taking into account the first and second derivatives with respect to the space variables and the first derivative in time. The method of asymptotic partial decomposition of the domain is justified for the time periodic problem.

The aim of the present study is to present an effect boundary element method (BEM) for electroelastic analysis of ultrathin piezoelectric films/coatings. The troublesome nearly singular integrals, which are crucial in applying the BEM for thin‐structural problems, are calculated accurately by using a nonlinear coordinate transformation method. The advanced BEM presented requires no remeshing procedure regardless of the thickness of the thin structure. Promising BEM results with only a small number of boundary elements can be achieved with the relative thickness of the thin piezoelectric film is as small as 10−8, which is sufficient for modeling many ultrathin piezoelectric films as used in smart materials and micro‐electro‐mechanical systems. The present BEM procedure with thin‐body capabilities is also extended to general multidomain problems and used to model ultrathin coating/substrate piezoelectric structures. The influence of relative layer‐to‐substrate thickness and the bimaterial mismatch parameters are carefully investigated. Excellent agreement between numerical and theoretical solutions has been demonstrated. [ABSTRACT FROM AUTHOR]

We prove norm-resolvent and spectral convergence in L² of solutions to the Neumann Poisson problem --Δε= f on a domain Ωε perforated by Dirichlet holes and shrinking to a 1- dimensional interval. The limit u satisfies an equation of the type --u" + μu = f on the interval (0, 1), where μ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighborhood and the vertex neighborhood is chosen correctly, the constant μ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation. [ABSTRACT FROM AUTHOR]

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order O (h 5) in the shell thickness h . The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. We give an explicit form of the curvature energy using the orthogonal Cartan-decomposition of the wryness tensor. Moreover, we consider the matrix representation of all tensors in the derivation of the variational formulation, because this is convenient when the problem of existence is considered, and it is also preferential for numerical simulations. The step by step construction allows us to give a transparent approximation of the three-dimensional parental problem. The resulting 6-parameter isotropic shell model combines membrane, bending and curvature effects at the same time. The Cosserat shell model naturally includes a frame of orthogonal directors, the last of which does not necessarily coincide with the normal of the surface. This rotation-field is coupled to the shell-deformation and augments the well-known Reissner-Mindlin kinematics (one independent director) with so-called in-plane drill rotations, the inclusion of which is decisive for subsequent numerical treatment and existence proofs. As a major novelty, we determine the constitutive coefficients of the Cosserat shell model in dependence on the geometry of the shell which are otherwise difficult to guess. [ABSTRACT FROM AUTHOR]

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including O (h 5) terms. The form of the energy allows us to show the coercivity for terms up to order O (h 5) and the convexity of the energy. Secondly, we consider only that part of the energy including O (h 3) terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including O (h 5) the conditions on the thickness h are those considered in the modelling process and they are independent of the constitutive parameter, in the O (h 3) -case the coercivity is proven under some more restrictive conditions on the thickness h . [ABSTRACT FROM AUTHOR]

A first order model for the transmission of waves through a sound-hard perforation along an interface is derived. Mathematically, we study the Neumann problem for the Helmholtz equation in a complex geometry, the domain contains a periodic array of inclusions of size ε > 0 along a co-dimension 1 manifold. We derive effective equations that describe the limit as ε → 0. At leading order, the Neumann sieve perforation has no effect. The corrector is given by a Helmholtz equation on the unperturbed domain with jump conditions across the interface. The corrector equations are derived with unfolding methods in L 1 -based spaces. [ABSTRACT FROM AUTHOR]

Powder based laser material deposition (LMD) is a suitable additive manufacturing technology for repair and overhaul of metal parts as well as for the build-up of new parts. This technology is used in industrial applications as automotive, turbomachinery, aviation, or tool making [T. Jambor, "Funktionalisierung von Bauteiloberflächen durch Mikro-Laserauftrag-schweißen," Ph.D. thesis, RWTH Aachen University, Germany, 2012; I. Kelbassa, "Qualifizieren des Laserstrahl-Auftragschweißens von BLISKs aus Nickel- und Titanbasislegierungen," Ph.D. thesis, RWTH Aachen University, Germany, 2006; J. Witzel, "Qualifizierung des Laserstrahl-Auftragschweißens zur generativen Fertigung von Luftfahrtkomponenten," Ph.D. thesis, RWTH Aachen University, Germany, 2014; R. Poprawe, Tailored Light 2 (Springer, Berlin, 2011), pp. 209–224]. The repair of turbine blade tips, for example, requires dimensional accuracy of the deposited volume to achieve sufficient overhang necessary for postweld machining and simultaneously reduces the time and efforts for machining to a minimum. Typically, the repair of a turbine blade tip is carried out layer by layer, and each layer starts with the deposition of a contour track. This contour track is important for outer geometry and therefore for the dimensional accuracy of the volume to be deposited. In order to determine the influence of the contour track on deposit geometry, a basic research in laser material deposition on edges of plates is carried out. During the research presented in this paper, the geometries of single tracks and webs deposited on edges are investigated depending on the distance of the track to the edge of the plate, on the lateral offset of the tracks of two successive layers, and the number of overlapping tracks in one layer. The melt pools are monitored by high-speed videography to analyze melt pool geometry and the characteristics of the powder grain trajectories of the filler metal. In the future, the results of this research will be used to validate a simulation tool that has been developed to simulate LMD on edges with the aim to reduce the number of experimental studies. [ABSTRACT FROM AUTHOR]

The time periodic Navier–Stokes equations are considered in the three-dimensional and two-dimensional settings with Dirichlet boundary conditions in thin tube structures. These structures are finite union of thin cylinders (thin rectangles in the case of dimension two), where the small parameter ε is the ratio of the hight and the diameter of the cylinders. We consider the case of finite or big coefficient before the time derivative. This setting is motivated by hemodynamical applications. Theorems of existence and uniqueness of a solution are proved. Complete asymptotic expansion of a solution is constructed and justified by estimates of the difference of the exact solution and truncated series of the expansion in norms taking into account the first and second derivatives with respect to the space variables and the first derivative in time. The method of asymptotic partial decomposition of the domain is justified for the time periodic problem. [ABSTRACT FROM AUTHOR]

We consider a spectral problem for the Laplacian operator in a planar T-like shaped thin structure Ω ε , where ε denotes the transversal thickness of both branches. We assume the homogeneous Dirichlet boundary condition on the ends of the branches and the homogeneous Neumann boundary condition on the remaining part of the boundary of Ω ε. We study the asymptotic behavior, as ε tends to zero, of the high frequencies of such a problem. Unlike the asymptotic behavior of the low frequencies where the limit problem involves only longitudinal vibrations along each branch of the T-like shaped thin structure (i.e. 1 D limit spectral problems), we obtain a two dimensional limit spectral problem which allows us to capture other kinds of vibrations. We also give a characterization of the asymptotic form of the eigenfunctions originating these vibrations [ABSTRACT FROM AUTHOR]

Level set–based optimization for two-dimensional structural configurations with thin members is presented. A structural domain with thin thickness is defined as a narrow band region on the zero-level contour of the level set function. No additional constraints or penalty functional is required to enforce semi-uniformity in member thickness. Design velocity is calculated on the zero level set, not on domain boundaries, and extended to level set grids in the narrow band. For complicated structural layouts, multiple level set functions are employed. The effectiveness of the proposed method is verified by solving optimization problems of bar configurations. Since no thickness constraints are employed, structurally unfavorable distorted joints seen in other literature do not appear in the results. [ABSTRACT FROM AUTHOR]

The method of asymptotic partial decomposition of a domain proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some special interface conditions between the three-dimensional and one-dimensional parts. In the case of fluid mechanics these conditions prescribe a precomputed Poiseuille-type shape of a solution at the interface, which, however, are not generalizable to the case with a boundary layer in time. In this work we present a new more general version of the method which considered and justified the transient Navier-Stokes equations. Although theoretical justification (well posedness, asymptotic analysis) can be shown only for moderate Reynolds numbers, the provided numerical tests show good accuracies for higher values. [ABSTRACT FROM AUTHOR]

Abstract We study the steady incompressible flow of a Bingham fluid in a thin T-like shaped domain, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. This phenomenon is described by non linear variational inequalities. By letting the parameter describing the thickness of the thin domain tend to zero, we derive two uncoupled problems corresponding to the two branches of the T-like shaped structure. We then analyze and give a physical justification of the limit problem. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a -limit with respect to an appropriate convergence, is defined in a 'stratified' Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for p-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectively. [ABSTRACT FROM AUTHOR]

Thin structures are often required for several engineering applications. Although thick sections are relatively easy to produce, the cutting of thin sections poses greater challenges, particularly in the case of thermal machining processes. The level of difficulty is increased if the thin sections are of larger lengths and heights. In this study, high-aspect-ratio thin structures of micrometer thickness (117–500 µm) were fabricated from D2 steel through wire electrical discharge machining. Machining conditions were kept constant, whereas the structure (fins) sizes were varied in terms of fin thickness (FT), fin height (FH), and fin length (FL). The effects of variation in FT, FH, and FL were assessed over the machining errors (FT and FL errors) and structure formation and its quality. Experiments were conducted in a phased manner (four phases) to determine the minimum possible FT and maximum possible FL that could be achieved without compromising the shape of the structure (straight and uniform cross-section). Thin structures of smaller lengths (1–2 mm long) can be fabricated easily, but, as the length exceeds 2 mm, the structure formation loses its shape integrity and the structure becomes broken, deflected, or deflected and merged at the apex point of the fins.

The description of the behavior of a material subjected to multi-physics loadings requires the formulation of constitutive laws that usually derive from Gibbs free energies, using invariant quantities depending on the considered physics and material symmetries. On the other hand, most of crystalline materials can be described by their crystalline texture and the associated preferred directions of strong crystalline symmetry (the so-called fibers). Moreover, among the materials produced industrially, many are manufactured in the form of sheets or of thin layers. This article has for object the study of the magneto-mechanical coupling which is a function of the stress σ and the magnetization M. We consider a material with cubic symmetry whose texture can be described by one of three fibers denoted as θ , γ or α ′ , and which is thin enough so that both the stress and the magnetization can be considered as in-plane quantities. We propose an algorithm able to derive linear relations between the 30 cubic invariants I k of a minimal integrity basis describing a magneto-elastic problem, when they are restricted to in-plane loading conditions and for different fiber orientations. The algorithm/program output is a reduced list of invariants of cardinal 7 for the {100}-oriented θ fiber, of cardinal 15 for the {110}-oriented α ′ fiber and of cardinal 8 for the {111}-oriented γ fiber. This reduction (compared to initial cardinal 30) can be of great help for the formulation of low-parameter macroscopic magneto-mechanical models. • We study the magneto-mechanical coupling for a thin material with cubic symmetry. • Three crystallographic orientations are considered: θ -fiber, α ′ -fiber and γ -fiber. • An algorithm is proposed to reduce sets of 3D invariants to sets of restricted (in-plane) invariants. • The corresponding generating sets for each fiber are significantly reduced. • An application to a quadratic in stress energy is given for the 3 considered fibers. [ABSTRACT FROM AUTHOR]

This article is devoted to the instrumentation, with optical fibres, of the straps holding the envelope of stratospheric balloons. This instrumentation is motivated in the first instance by the need to validate the numerical models used in the design of balloons. It must also be used to measure the temperature along the envelope in order to deduce the pressure field. It is shown at first that the optical fibres can be inserted inside a strap during its fabrication. Different kinds of insertion are considered, none of them perturb the industrial process. The instrumented straps were then submitted to thermal and mechanical tests and the distributed Brillouin frequency shifts were measured. We thus determined the type of insertion to be used according to the parameter (temperature or strain) to be measured and assessed the performance of the measurement chain.

Highlights • Introducing our open-source FEniCS-Shells library that is freely available to the community. • Providing a novel interpretation and implementation of the MITC discretisation for plate and shells. • Improving and extending to nonlinear shells a PSRI approach initially introduced for linear shells by Arnold and Brezzi. • Proposing new verification tests for weakly nonlinear shell models. Abstract A large number of advanced finite element shell formulations have been developed, but their adoption is hindered by complexities of transforming mathematical formulations into computer code. Furthermore, it is often not straightforward to adapt existing implementations to emerging frontier problems in thin structural mechanics including nonlinear material behaviour, complex microstructures, multi-physical couplings, or active materials. We show that by using a high-level mathematical modelling strategy and automatic code generation tools, a wide range of advanced plate and shell finite element models can be generated easily and efficiently, including: the linear and non-linear geometrically exact Naghdi shell models, the Marguerre-von Kármán shallow shell model, and the Reissner-Mindlin plate model. To solve shear and membrane-locking issues, we use: a novel re-interpretation of the Mixed Interpolation of Tensorial Component (MITC) procedure as a mixed-hybridisable finite element method, and a high polynomial order Partial Selective Reduced Integration (PSRI) method. The effectiveness of these approaches and the ease of writing solvers is illustrated through a large set of verification tests and demo codes, collected in an open-source library, FEniCS-Shells, that extends the FEniCS Project finite element problem solving environment. [ABSTRACT FROM AUTHOR]

Given a distribution of defects on a structured surface, such as those represented by 2-dimensional crystalline materials, liquid crystalline surfaces, and thin sandwiched shells, what is the resulting stress field and the deformed shape? Motivated by this concern, we first classify, and quantify, the translational, rotational, and metrical defects allowable over a broad class of structured surfaces. With an appropriate notion of strain, the defect densities are then shown to appear as sources of strain incompatibility. The strain incompatibility relations, aided with a decomposition of strain into elastic and plastic parts, and the stress equilibrium relations, with a suitable choice of material response, provide the necessary equations for determining both the stress field and the deformed shape. We demonstrate this by applying our theory to Kirchhoff-Love shells with a kinematics which allows for small surface strains but moderately large rotations. We discuss implications of our framework in the context of 2-dimensional crystals, growing biological membranes, and isotropic fluid films. [ABSTRACT FROM AUTHOR]

The analysis of thin curvilinear objects in 3D images is a complex and challenging task. In this article, we introduce a new, non-linear operator, called RORPO (Ranking the Orientation Responses of Path Operators). Inspired by the multidirectional paradigm currently used in linear filtering for thin structure analysis, RORPO is built upon the notion of path operator from mathematical morphology. This operator, unlike most operators commonly used for 3D curvilinear structure analysis, is discrete, non-linear and non-local. From this new operator, two main curvilinear structure characteristics can be estimated: an intensity feature, that can be assimilated to a quantitative measure of curvilinearity; and a directional feature, providing a quantitative measure of the structure's orientation. We provide a full description of the structural and algorithmic details for computing these two features from RORPO, and we discuss computational issues. We experimentally assess RORPO by comparison with three of the most popular curvilinear structure analysis filters, namely Frangi Vesselness, Optimally Oriented Flux, and Hybrid Diffusion with Continuous Switch. In particular, we show that our method provides up to 8 percent more true positive and 50 percent less false positives than the next best method, on synthetic and real 3D images. [ABSTRACT FROM PUBLISHER]

We consider graphs parameterized on a portion X ⊂ Z d × { 1 , ... , M } k of a cylindrical subset of the lattice Z d × Z k , and perform a discrete-to-continuum dimension-reduction process for energies defined on X of quadratic type. Our only assumptions are that X be connected as a graph and periodic in the first d -directions. We show that, upon scaling of the domain and of the energies by a small parameter ɛ , the scaled energies converge to a d -dimensional limit energy. The main technical points are a dimension-reducing coarse-graining process and a discrete version of the p -connectedness approach by Zhikov. [ABSTRACT FROM AUTHOR]

Article represents the study of applied problems of mathematics, whose mathematical modeling leads to boundary problems for equations in partial derivatives. Mathematical methods, applied to these models, enable to obtain exact analytical results. Detailed result is represented for boundary problem of oscillations of thin structures with boundary conditions in general terms. Application of spectral decomposition for sufficiently smooth function, characterizing the membrane deviation from equilibrium state, enables to define exact analytic representation of inflection function for studied problem. To calculate multilayer plates, method of finite elements is applied.

The problem of design optimization for heterogeneous plates which inherit a periodic pattern is studied under the constraint of anisotropy based on the effective bending stiffness model on a unit periodic cell obtained by a homogenization technique. [ABSTRACT FROM AUTHOR]

We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on deformations of the beam. In particular, we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise regular curves representing the midline of the beam. The elastic part of this functional is the classical Euler-Bernoulli functional for thin beams in the bending dominated regime in terms of the curve's curvature. In addition there also emerges a fracture term proportional to the number of discontinuities of the curve and its first derivative. [ABSTRACT FROM AUTHOR]

To improve the far-field directivity properties of a given mid-range acoustic horn, previously designed by shape optimization to exhibit almost ideal transmission properties in the frequency range 1.6-9.05 kHz, we apply layout optimization of thin sound-hard material in the interior of the horn. The purpose of the optimization is to place scattering material to prevent the sound intensity to increasingly be concentrated, with increasing frequency, along the horn axis. Absence or presence of thin sound-hard material is modeled by an equivalent surface transmission impedance, and the optimization algorithm determines the distribution of air or sound-hard material along a 'ground structure' in the form of a grid inside the horn. The surface impedance is numerically handled using a newly developed finite-element formulation that allows exact enforcement of a vanishing impedance, corresponding to air, which would not be possible using a standard formulation. Horns provided with the optimized scatterers show a much improved angular coverage, compared to the initial configuration, with beam widths that exceed 60 uniformly over the operational frequency range, without destroying the good transmission properties of the initial horn. [ABSTRACT FROM AUTHOR]

In this paper, two quadratic solid-shell (SHB) elements are proposed for the three-dimensional modeling of thin structures. These consist of a 20-node hexahedral solid-shell element, denoted SHB20, and its 15-node prismatic counterpart, denoted SHB15. The formulation of these elements is extended in this work to include geometric and material nonlinearities, for application to problems involving large displacements and rotations as well as plasticity. For this purpose, the SHB elements are coupled with large-strain anisotropic elasto-plastic constitutive equations for metallic materials. Although based on a purely three-dimensional approach, several modifications are introduced in the formulation of these elements to provide them with interesting shell features. In particular, a special direction is chosen to represent the thickness, along which a user-defined number of integration points are located. Furthermore, for efficiency requirements and for alleviating locking phenomena, an in-plane reduced-integration scheme is adopted. The resulting formulations are implemented into the finite element software ABAQUS/Standard and, to assess their performance, a variety of nonlinear benchmark problems are investigated. Attention is then focused on the simulation of various complex sheet metal forming processes, involving large strain, anisotropic plasticity, and double-sided contact. From all simulation results, it appears that the SHB elements represent an interesting alternative to traditional shell and solid elements, due to their versatility and capability of accurately modeling selective nonlinear benchmark problems as well as complex sheet metal forming processes. [ABSTRACT FROM AUTHOR]

We minimize failure criteria with respect to element-wise fiber orientation in laminae (and laminates) undergoing finite strains. A fully mechanical optimization approach is adopted: the analysis is encapsulated in a Newmark time integration equivalent to Nesterov's first-order minimization algorithm, with Newton iteration followed by the solution of an adjoint system to obtain analytical sensitivities. This is implemented in our in-house software, Simplas. We consider transversely isotropic elasticity; and hyperelasticity, via the homogenized model of an incompressible Neo-Hookean material filled with cylindrical (fiber-like) pores. Two stress-based criteria are adopted: Tsai–Wu and modified Tsai–Hill. A finite strain solid-shell element, known to be locking-free, is used and here extended to perform the sensitivity operations. Three examples of optimized transversely isotropic elastic composites are shown, exhibiting remarkable advantages when compared with traditional optimization algorithms. One example is dedicated to finding optimal cylindrical void orientation in a hyperelasticity framework. The algorithm works for very high values of deformation. During stress minimization, stiffness can either decrease or increase, and this was observed in the numerical experiments. • Equivalent stress minimization is performed in laminate shells. • Optimal fiber/void orientation. • Finite strains and hyperelasticity are considered. • Time-integrated version of Nesterov's algorithm is adopted. • Benchmarking is performed with respect to published results. [ABSTRACT FROM AUTHOR]

In this work a three dimensional (3D) boundary element method was established with an efficient nonlinear coordinate transformation scheme, namely sinh transformation, to evaluate nearly singular integrals in boundary integral formulations. Second-order quadrilateral surface elements were developed based on this method to accurately describe the geometry of thin structures. The elastic behaviors of selected thin structures were then computed by using the 3D boundary element model to demonstrate the accuracy and efficiency of this approach. A number of testing examples, i.e., the 3D Kirsch problem, the thin spherical shell problem, the ellipsoidal vessel problem with non-uniform thickness and the hollow circular cylinder problem, were numerically studied to test the established method. Remarkable accuracy and efficiency were found in the developed approach through the comparison to the numerical results and analytical solutions reported in the literature. [ABSTRACT FROM AUTHOR]

In the current contribution, prismatic and hexahedral quadratic solid–shell (SHB) finite elements are proposed for the geometrically nonlinear analysis of thin structures made of functionally graded material (FGM). The proposed SHB finite elements are developed within a purely 3D framework, with displacements as the only degrees of freedom. Also, the in-plane reduced-integration technique is combined with the assumed-strain method to alleviate various locking phenomena. Furthermore, an arbitrary number of integration points are placed along a special direction, which represents the thickness. The developed elements are coupled with functionally graded behavior for the modeling of thin FGM plates. To this end, the Young modulus of the FGM plate is assumed to vary gradually in the thickness direction, according to a volume fraction distribution. The resulting formulations are implemented into the quasi-static ABAQUS/Standard finite element software in the framework of large displacements and rotations. Popular nonlinear benchmark problems are considered to assess the performance and accuracy of the proposed SHB elements. Comparisons with reference solutions from the literature demonstrate the good capabilities of the developed SHB elements for the 3D simulation of thin FGM plates.

Morphoelastic rods are thin bodies which can grow and can change their intrinsic curvature and torsion. We deduce a system of equations that rule accretion and remodeling in a morphoelastic rod by combining balance laws involving non-standard forces with constitutive prescriptions filtered by a dissipation principle that takes into account both standard and non-standard working. We find that, as in the theory of three-dimensional bulk growth proposed [DiCarlo, A and Quiligotti, S. Mech Res Commun 2002; 29: 449–456], it is possible to identify a universal coupling mechanism between stress and growth, conveyed by an Eshelbian driving force. [ABSTRACT FROM AUTHOR]

An immersed finite element fluid-structure interaction algorithm with an anisotropic remeshing strategy for thin rigid structures is presented in two dimensions. One specific feature of the algorithm consists of remeshing only the fluid elements that are cut by the solid such that they fit the solid geometry. This approach allows to keep the initial (given) fluid mesh during the entire simulation while remeshing is performed locally. Furthermore, constraints between the fluid and the solid may be directly enforced with both an essential treatment and elements allowing the stress to be discontinuous across the structure. Remeshed elements may be strongly anisotropic. Classical interpolation schemes - inf-sup stable on isotropic meshes - may be unstable on anisotropic ones. We specifically focus on a proper finite element pair choice. As for the time advancing of the fluid-structure interaction solver, we perform a geometrical linearization with a sequential solution of fluid and structure in a backward Euler framework. Using the proposed methodology, we extensively address the motion of a hinged rigid leaflet. Numerical tests demonstrate that some finite element pairs are inf-sup unstable with our algorithm, in particular with a discontinuous pressure. [ABSTRACT FROM AUTHOR]

This paper presents a fully coupled thermo-mechanical constitutive model for shape memory alloys taking into account latent heat effects during forward and reverse martensitic phase transformations. This model is obtained as an extension of an original macroscopic model, derived from a micromechanical-inspired Gibbs free energy expression. The proposed formulation considers beside the constitutive equations describing the thermo-mechanical behavior of Shape Memory Alloys (SMAs), the heat equation with additional internal source terms related to the latent heat generated during phase transformation. Hence, the thermo-mechanical problem to be solved consists of the mechanical equilibrium and the established heat equations. A 2D specific plane stress continuum and isoparametric finite element with three degrees of freedom (dof) per node corresponding to in-plane displacements and temperature is developed to solve the derived thermomechanical problem. The developed 2D finite element is implemented in the Abaqus ® finite element code through the UEL user’s subroutine. The numerical simulations performed by using this new finite element show the delaying effect of the transformation latent heat on the forward and reverse phase transformations in SMA thin structures and the possible heterogeneous character of the phase transformation in this case. These effects are even more important as the strain rate is high. [ABSTRACT FROM AUTHOR]

This work looks at the asymptotic behaviour of the solution to the Helmholtz equation in a penetrable domain of $\mathbb{R}$3 with a thin layer of thickness δ which tends to 0. We use the method of multi-scale expansion to derive and justify an asymptotic expansion of the solution with respect to the thickness δ up to any order. We then provide approximate transmission conditions of order two defined on an interface located inside the thin layer, with accuracy up to O(δ2), which allow one to take into account the influence of the thin layer. [ABSTRACT FROM PUBLISHER]

We present a variational model to perform the segmentation of thin structures in MRI images (namely codimension 1 objects). It is based on the classical Mumford-Shah functional and we have added geometrical priors as constraints. We precisely describe the structure model (that we call tubes). We give existence, uniqueness and regularity results for the solution to the optimization problem. The keypoint is the fact that 2D/3D problems are equivalent to 1D ones. This gives hints to perform an automatic parameter tuning for numerical purpose. [ABSTRACT FROM AUTHOR]

This paper is concerned with the vibration isolation efficiency analysis of total or partially buried thin walled wave barriers in poroelastic soils. A two-dimensional time harmonic model that treats soils and structures in a direct way by combining appropriately the conventional Boundary Element Method (BEM), the Dual BEM (DBEM) and the Finite Element Method (FEM) is developed to this aim. The wave barriers are impinged by Rayleigh waves obtained from Biot’s poroelasticity equations assuming a permeable free-surface. The suitability of the proposed model is justified by comparison with available previous results. The vibration isolation efficiency of three kinds of wave barriers (open trench, simple wall, open trench-wall) in poroelastic soils is studied by varying their geometry, the soil properties and the frequency. It is found that the efficiency of these wave barriers behaves similarly to these in elastic soils, except for high porosities and small dissipation coefficients. The efficiency of open trench-wall barriers can be evaluated neglecting their walls if they are typical sheet piles. This does not happen with walls of bigger cross-sections, leading in general to efficiency losses. Likewise, increasing the burial depth to trench depth ratio has a negative impact on the efficiency. [ABSTRACT FROM AUTHOR]

In this paper, a framework for computational homogenization of shell structures is proposed in the context of small-strain elastostatics, with extensions to large displacements and large rotations. At the macroscopic scale, heterogeneous thin structures are modeled using a homogenized shell model, based on a versatile three-dimensional seven-parameter shell formulation, incorporating a through-thickness and pre-integrated constitutive relationship. In the context of small strains, we show that the local solution on the elementary cell can be decomposed into six strains and six-strain gradient modes, associated with corresponding boundary conditions. The heterogeneities can have arbitrary morphology but are assumed to be periodically distributed in the tangential direction of the shell. We then propose an extension of the small-strain framework to geometrical nonlinearities. The procedure is purely sequential and does not involve coupling between scales. The homogenization method is validated and illustrated through examples involving large displacements and buckling of heterogeneous plates and shells. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider a residually stressed plate-like body having the shape of a cylinder of cross-section ω and thickness hε, subjected to a system of loads proportional to a positive multiplier λ. We look for the smallest value of the multiplier such that the plate buckles, the so-called critical multiplier. The critical multiplier is computed by minimizing a functional whose domain of definition is a collection of vector fields defined in the three-dimensional region Ωε =ω ×(−εh/2,+εh/2). We let ε → 0 and we show that if the residual stress and the incremental stress induced by the applied loads scale with ε in a suitable manner, then the critical multiplier converges to a limit that can be computed by minimizing a functional whose domain is a collection of scalar fields defined on the two-dimensional region ω. In the special case of null residual stress, the Euler–Lagrange equations associated to this functional coincide with the classical equations governing plate buckling. [ABSTRACT FROM AUTHOR]