Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks

In this paper a novel device aimed at controlling the mechanical vibrations of plates by means of a set of electrically-interconnected piezoelectric actuators is described. The actuators are embedded uniformly in the plate wherein they connect every node of an electric network to ground, thus playing the two-fold role of capacitive element in the electric network and of couple suppliers. A mathematical model is introduced to describe the propagation of electro-mechanical waves in the device; its validity is restricted to the case of wave-forms with wave-length greater than the dimension of the piezoelectric actuators used. A self-resonance criterion is established which assures the possibility of electro-mechanical energy exchange. Finally the problem of vibration control in simply supported and clamped plates is addressed; the optimal net-impedance is determined. The results indicate that the proposed device can improve the performances of piezoelectric actuationComment: 22 page

Concurrently coupled solid shell-based adaptive multiscale method for fracture

Artículo Open Access en el sitio web del editor. Pago por publicar en abierto.A solid shell-based adaptive atomistic–continuum numerical method is herein proposed to simulate complex crack growth patterns in thin-walled structures. A hybrid solid shell formulation relying on the combined use of the enhanced assumed strain (EAS) and the assumed natural strain (ANS) methods has been considered to efficiently model the material in thin structures at the continuum level. The phantom node method (PNM) is employed to model the discontinuities in the bulk. The discontinuous solid shell element is then concurrently coupled with a molecular statics model placed around the crack tip. The coupling between the coarse scale and the fine scale is realized through the use of ghost atoms, whose positions are interpolated from the coarse scale solution and enforced as boundary conditions to the fine scale model. In the proposed numerical scheme, the fine scale region is adaptively enlarged as the crack propagates and the region behind the crack tip is adaptively coarsened in order to reduce the computation costs. An energy criterion is used to detect the crack tip location. All the atomistic simulations are carried out using the LAMMPS software. A computational framework has been developed in MATLAB to trigger LAMMPS through system command. This allows a two way interaction between the coarse and fine scales in MATLAB platform, where the boundary conditions to the fine region are extracted from the coarse scale, and the crack tip location from the atomistic model is transferred back to the continuum scale. The developed framework has been applied to study crack growth in the energy minimization problems. Inspired by the influence of fracture on current–voltage characteristics of thin Silicon photovoltaic cells, the cubic diamond lattice structure of Silicon is used to model the material in the fine scale region, whilst the Tersoff potential function is employed to model the atom–atom interactions. The versatility and robustness of the proposed methodology is demonstrated by means of several fracture applications.Unión Europea ERC 306622Ministerio de Economía y Competitividad DPI2012-37187, MAT2015-71036-P y MAT2015-71309-PJunta de Andalucía P11-TEP-7093 y P12-TEP -105

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis

In this dissertation, we present a methodology for understanding the propagation and control of geometric variation in engineering design and analysis. This work is comprised of two major components: (i) novel discretizations and associated solution strategies for rapid numerical solution over geometric parametrizations of the linear and nonlinear thin-shell equations, and (ii) efficient surrogate modeling techniques and algorithms towards the control of geometric variation. While the methodologies presented are in the setting of structural mechanics, particularly Nitsche's method in the context of linearized membranes, Kirchhoff-Love plates, and Kirchhoff-Love shells, they are applicable to any system of parametric partial differential equations. We present a design space exploration framework that elucidates design parameter sensitivities used to inform initial and early-stage design and a novel tolerance allocation algorithm for the assessment and control of geometric variation on system performance. Both of these methodologies rely on surrogate modeling techniques where various designs throughout the design space considered are sampled and used in the construction of approximations to the system response. The design space exploration paradigm enables the visualization of a full system response through the surrogate model approximation. The tolerance allocation algorithm poses a set of optimization problems over this surrogate model restricted to nested hyperrectangles represents the effect of prescribing design tolerances, where the maximizer of this restricted function depicts the worst-case member, i.e. design. The loci of these tolerance hyperrectangles with maximizers attaining the performance constraint represents the boundary to the feasible region of allocatable tolerances. The boundary of the feasible set is elucidated as an immersed manifold of codimension one, over which optimization routines exist and are employed to efficiently determine an optimal feasible tolerance with respect to a user-specified measure. Examples of these methodologies for problems of various complexities are presented

Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis

In this dissertation, we present a methodology for understanding the propagation and control of geometric variation in engineering design and analysis. This work is comprised of two major components: (i) novel discretizations and associated solution strategies for rapid numerical solution over geometric parametrizations of the linear and nonlinear thin-shell equations, and (ii) efficient surrogate modeling techniques and algorithms towards the control of geometric variation. While the methodologies presented are in the setting of structural mechanics, particularly Nitsche's method in the context of linearized membranes, Kirchhoff-Love plates, and Kirchhoff-Love shells, they are applicable to any system of parametric partial differential equations. We present a design space exploration framework that elucidates design parameter sensitivities used to inform initial and early-stage design and a novel tolerance allocation algorithm for the assessment and control of geometric variation on system performance. Both of these methodologies rely on surrogate modeling techniques where various designs throughout the design space considered are sampled and used in the construction of approximations to the system response. The design space exploration paradigm enables the visualization of a full system response through the surrogate model approximation. The tolerance allocation algorithm poses a set of optimization problems over this surrogate model restricted to nested hyperrectangles represents the effect of prescribing design tolerances, where the maximizer of this restricted function depicts the worst-case member, i.e. design. The loci of these tolerance hyperrectangles with maximizers attaining the performance constraint represents the boundary to the feasible region of allocatable tolerances. The boundary of the feasible set is elucidated as an immersed manifold of codimension one, over which optimization routines exist and are employed to efficiently determine an optimal feasible tolerance with respect to a user-specified measure. Examples of these methodologies for problems of various complexities are presented

Smooth representation of thin shells and volume structures for isogeometric analysis

The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part. First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of “isoparametric”, for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images. Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids. Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1 continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems

A rapid and efficient isogeometric design space exploration framework with application to structural mechanics

In this paper, we present an isogeometric analysis framework for design space exploration. While the methodology is presented in the setting of structural mechanics, it is applicable to any system of parametric partial differential equations. The design space exploration framework elucidates design parameter sensitivities used to inform initial and early-stage design. Moreover, this framework enables the visualization of a full system response, including the displacement and stress fields throughout the domain, by providing an approximation to the system solution vector. This is accomplished through a collocation-like approach where various geometries throughout the design space under consideration are sampled. The sampling scheme follows a quadrature rule while the physical solutions to these sampled geometries are obtained through an isogeometric method. A surrogate model to the design space solution manifold is constructed through either an interpolating polynomial or pseudospectral expansion. Examples of this framework are presented with applications to the Scordelis–Lo roof, a Flat L-Bracket, and an NREL 5 MW wind turbine blade