NURBS-enhanced finite element method (NEFEM) Sevilla, Ruben; Fernández-Méndez, Sonia; Huerta, Antonio
International journal for numerical methods in engineering,
1 October 2008, Letnik:
76, Številka:
1
Journal Article, Publication
An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing ...the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.
•Thermal frequencies of MWCNT sandwich are computed using the HSDT and FE steps.•The accuracy of FE predictions is verified with the experimental frequencies.•The numerical results are obtained via ...experimental properties of the sandwich.•Microcontroller thermal chamber is fabricated and utilized for the experimentation.
The modal responses of multi-walled carbon nanotube-reinforced composite sandwich structural plate are computed under the elevated temperature environment using a higher-order polynomial kinematic model and the isoparametric finite element steps. The proposed model accuracy has been verified with experimental modal values under the influence of elevated temperature and ambient conditions. To perform the modal experiment the nanotube-reinforced composite sandwich panel filled with the epoxy core is fabricated. Further, the experimental elastic properties of epoxy, nanotube/epoxy composite and the sandwich are obtained individually for the current computational purpose. A tailor-made finite element computer code (MATLAB environment) is prepared using the multiscale mathematical formulation for the evaluation of thermal frequencies of the nanotube sandwich panel. The impact type vibration analyser has been utilized for the current testing purpose with the help of three components i.e. hardware (compact data acquisition system, cDAQ-9178), software (LAB-VIEW) and a microcontroller controlled thermal chamber (to maintain temperature profile). Finally, wide varieties of numerical examples are solved using the proposed computational model for the different design-related parameters. The intrinsic behaviour of each parameter on the epoxy-filled nanotube sandwich construction including the elevated temperature loading is discussed in details.
We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension ...of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.
In this study, we present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear ...elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of Rd. Our AMLMC algorithm is built on the results of the weak convergence rates in the work (Moon et al., 2006) for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The above-mentioned deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. In particular, for a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. Furthermore, we discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation.
From the numerical experiments, based on a Fourier expansion of the diffusivity coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo (MC) and standard MLMC (SMLMC) for a problem with a singularity similar to that at the tip of a slit modeling a crack.
•The single layer type HSDT kinematic applicability is verified for skew sandwich.•The FE eigenvalues are predicted considering the drilling degrees of freedom.•The numerical model accuracy is ...verified with experimental comparisons.•The fabricated specimen experimental properties are implemented for the analysis.•The inevitability of HSDT type model is established from the current analysis.
The experimental eigenvalue responses of the epoxy-filled skew sandwich structure are computed first-time in this research to show the suitability of equivalent type single-layer higher-order theory (including through-thickness stretching term effect) for the analysis. The sandwich shell model is formulated mathematically for the variable geometrical configurations considering the effect of skew angles. Further, the motion equation of the vibrated structure is solved numerically with the advent of the linear isoparametric finite element technique. Firstly, the numerical solution accuracy is established by verifying the modal values with the already published data including their element sensitivity test. Also, a few experimental frequencies (first-five mode) are recorded (impact type vibration analyzer) using the in-house fabricated sandwich plate components considering the experimental material properties for the comparison purpose. Moreover, a simulation model is prepared using the commercial package to show the efficacy of the presently proposed single layer theory for the analysis of the sandwich structure with and without the skew angle effect. The present comparison indicates that the proposed equivalent single-layer model is capable of solving the modal responses considering different structural input parameters (number and stacking sequences of the face sheet layers and variable aspect/thickness ratios) with adequate accuracy. Lastly, the verified model is explored to show its applicability by solving different numerical examples due to the change in their basic input parameters affecting the geometry, material properties and the stiffness.
The paper presents a unified approach for the a posteriori generation of arbitrary high-order curvilinear meshes via a solid mechanics analogy. The approach encompasses a variety of methodologies, ...ranging from the popular incremental linear elastic approach to very sophisticated non-linear elasticity. In addition, an intermediate consistent incrementally linearised approach is also presented and applied for the first time in this context. Utilising a consistent derivation from energy principles, a theoretical comparison of the various approaches is presented which enables a detailed discussion regarding the material characterisation (calibration) employed for the different solid mechanics formulations. Five independent quality measures are proposed and their relations with existing quality indicators, used in the context of a posteriori mesh generation, are discussed. Finally, a comprehensive range of numerical examples, both in two and three dimensions, including challenging geometries of interest to the solids, fluids and electromagnetics communities, are shown in order to illustrate and thoroughly compare the performance of the different methodologies. This comparison considers the influence of material parameters and number of load increments on the quality of the generated high-order mesh, overall computational cost and, crucially, the approximation properties of the resulting mesh when considering an isoparametric finite element formulation.
The conventional representation of isotropic hyperelastic strain energy densities as functions of scalar invariants of finite deformation tensors does not naturally extend to the field of anisotropic ...mechanics. Formulating an invariant-free representation of the strain energy function, fourth-order Orthotropic Lamé tensors define the constitutive law whilst naturally collapsing to the transversely isotropic and fully isotropic case where necessary simply as a by-product of known material symmetries. In this study, a simple linear isoparametric hexahedral finite element capable of describing anisotropic invariant-free hyperelasticity is presented. Careful conversion of the fourth-order tensor operations present in the strain energy function to computational arrays then applying to the principle of virtual work generates a weak formulation for finite element analyses. The finite element is then applicable to materials of any degree of anisotropy or compressibility and is particularly useful for predicting highly nonlinear responses such as the stiffening of fibrous biological tissues. A discussion of simple shear experimentation and modelling follows, as well as remarks on modelling nearly-incompressible materials.
A Ciarlet-Raviart type isoparametric mixed finite element method (MFEM) is constructed and analyzed for solving a class of fourth-order elliptic equation with Navier boundary condition defined on a ...curved domain in R2, and numerical quadrature is also considered in the scheme. The existence and uniqueness of the numerical solutions are proved under certain numerical quadrature. With the help of the special technically analysis, the optimal error estimates with H1 norm are obtained in Ωh, which yields better accuracy than using a convex polygonal domain to approximate the curved domain. For either constant coefficients or nonconstant coefficients problem, numerical examples are listed to confirm theoretical analysis respectively.
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