729 research outputs found
A virtual element method for the vibration problem of Kirchhoff plates
The aim of this paper is to develop a virtual element method (VEM) for the
vibration problem of thin plates on polygonal meshes. We consider a variational
formulation relying only on the transverse displacement of the plate and
propose an conforming discretization by means of the VEM which is
simple in terms of degrees of freedom and coding aspects. Under standard
assumptions on the computational domain, we establish that the resulting
schemeprovides a correct approximation of the spectrum and prove optimal order
error estimates for the eigenfunctions and a double order for the eigenvalues.
The analysis restricts to simply connected polygonal clamped plates, not
necessarily convex. Finally, we report several numerical experiments
illustrating the behaviour of the proposed scheme and confirming our
theoretical results on different families of meshes. Additional examples of
cases not covered by our theory are also presented
The singular dynamic method for dynamic contact of thin elastic structures
This paper adresses the approximation of the dynamic impact of thin elastic structures. The principle of the presented method is the use of a singular mass matrix obtained by different discretizations of the deflection and velocity. The obtained semi-discretized problem is proved to be well-posed and energy conserving. The method is applied on some membrane, beam and plate models and associated numerical experiments are discussed
Mixed finite elements for Kirchhoff-Love plate bending
We present a mixed finite element method with parallelogram meshes for the
Kirchhoff-Love plate bending model. Critical ingredient is the construction of
appropriate basis functions that are conforming in terms of a sufficiently
large tensor space and allow for any kind of physically relevant Dirichlet and
Neumann boundary conditions. For Dirichlet boundary conditions, and polygonal
convex or non-convex plates that can be discretized by parallelogram meshes, we
prove quasi-optimal convergence of the mixed scheme. Numerical results for
regular and singular examples with different boundary conditions illustrate our
findings.Comment: corrected versio
Blast analysis of enclosure masonry walls using homogenization approaches
A simple rigid-plastic homogenization model for the analysis of enclosure masonry walls sub- jected to blast loads is presented. The model is characterized by a few material parameters, is numerically inexpensive and very stable, and allows full parametric studies of entire walls subject to blast pressures. With the aim of considering the actual brickwork strength along vertical and horizontal axes, masonry out-of-plane anisotropic failure surfaces are obtained by means of a compatible homogenized limit analysis approach. In the model, a 3D system of rigid infinitely strong bricks connected by joints reduced to interfaces is identified with a 2D Kirchhoff-Love plate. For the joints, which obey an associated flow rule, aMohr-Coulomb fail- ure criterion with a tension cutoff and a linearized elliptic compressive cap is considered. In this way, the macroscopic masonry failure surface is obtained as a function of the macroscopic bending, torque, and in-plane forces by means of a linear programming problem in which the internal power dissipated is minimized. Triangular Kirchhoff-Love elements with linear in- terpolation of the displacements field and constant moment within each element are used at a structural level. In this framework, a simple quadratic programming problem is obtained to analyze entire walls subjected to blast loads. The multiscale strategy presented is adopted to predict the behavior of a rectangular wall supported on three sides (left, bottom, and right) representing an envelope wall in a building and subjected to a standardized blast load. The top edge of the wall is assumed unconstrained due to an imperfect connection (often an inter- layer material is used to prevent damage in the in-fill wall). A comparison with a standard elastic-plastic heterogeneous 3D analysis conducted with a commercial FE code is also pro- vided for a preliminary verification of the procedure at a structural level. The good agreement found and the very limited computational effort required for the simulations conducted with the presented model indicate that the proposed simple tool can be used by practitioners for the safety assessment of out-of-plane loaded masonry panels subjected to blast loading. An ex- haustive parametric analysis is finally conducted with different wall thicknesses, joint tensile strengths, and dynamic pressures, corresponding to blast loads (in kilograms of TNT) ranging from small to large
Automated shape and thickness optimization for non-matching isogeometric shells using free-form deformation
Isogeometric analysis (IGA) has emerged as a promising approach in the field
of structural optimization, benefiting from the seamless integration between
the computer-aided design (CAD) geometry and the analysis model by employing
non-uniform rational B-splines (NURBS) as basis functions. However, structural
optimization for real-world CAD geometries consisting of multiple non-matching
NURBS patches remains a challenging task. In this work, we propose a unified
formulation for shape and thickness optimization of separately-parametrized
shell structures by adopting the free-form deformation (FFD) technique, so that
continuity with respect to design variables is preserved at patch intersections
during optimization. Shell patches are modeled with isogeometric
Kirchhoff--Love theory and coupled using a penalty-based method in the
analysis. We use Lagrange extraction to link the control points associated with
the B-spline FFD block and shell patches, and we perform IGA using the same
extraction matrices by taking advantage of existing finite element assembly
procedures in the FEniCS partial differential equation (PDE) solution library.
Moreover, we enable automated analytical derivative computation by leveraging
advanced code generation in FEniCS, thereby facilitating efficient
gradient-based optimization algorithms. The framework is validated using a
collection of benchmark problems, demonstrating its applications to shape and
thickness optimization of aircraft wings with complex shell layouts
Powell-Sabin B-splines and unstructured standard T-splines for the solution of the Kirchhoff-Love plate theory exploiting BĂ©zier extraction
The equations that govern Kirchhoff–Love plate theory are solved using quadratic Powell–Sabin B-splines and unstructured standard T-splines. Bézier extraction is exploited to make the formulation computationally efficient. Because quadratic Powell–Sabin B-splines result in inline image-continuous shape functions, they are of sufficiently high continuity to capture Kirchhoff–Love plate theory when cast in a weak form. Unlike non-uniform rational B-splines (NURBS), which are commonly used in isogeometric analysis, Powell–Sabin B-splines do not necessarily capture the geometry exactly. However, the fact that they are defined on triangles instead of on quadrilaterals increases their flexibility in meshing and can make them competitive with respect to NURBS, as no bending strip method for joined NURBS patches is needed. This paper further illustrates how unstructured T-splines can be modified such that they are inline image-continuous around extraordinary points, and that the blending functions fulfil the partition of unity property. The performance of quadratic NURBS, unstructured T-splines, Powell–Sabin B-splines and NURBS-to-NURPS (non-uniform rational Powell–Sabin B-splines, which are obtained by a transformation from a NURBS patch) is compared in a study of a circular plat
Lower bound static approach for the yield design of thick plates
International audienceThe present work addresses the lower bound limit analysis (or yield design) of thick plates under shear-bending interaction. Equilibrium finite elements are used to discretize the bending moment and the shear force fields. Different strength criteria, formulated in the five-dimensional space of bending moment and shear force, are considered, one of them taking into account the interaction between bending and shear resistances. The criteria are chosen to be sufficiently simple so that the resulting optimization problem can be formulated as a second-order cone programming problem, which is solved by the dedicated solver MOSEK. The efficiency of the proposed finite element is illustrated by means of numerical examples on different plate geometries, for which the thin plate solutions as well as the pure shear solutions are accurately obtained as two different limit cases of the plate slenderness ratio. In particular, the proposed element exhibits a good behavior in the thin plate limit
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