Estimates of the modeling error for the Kirchhoff–Love plate model

Repin, Sergey; Sauter, Stéfan A.

doi:10.1016/j.crma.2010.09.004

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…for problems in elasticity, homogenization, and dimension reduction, see e.g. [4,5,11,12,20,30,31,32,36,37,38,40,41,42]. However, the field of a posteriori modeling error estimates is far from being fully developed; in fluid mechanics, few results on a posteriori modeling error estimates have been available so far, amongst them the error estimate for the model simplification of replacing the stationary incompressible Navier-Stokes equation with the Stokes equation [30] as well as the modeling error estimate for certain model simplifications in combustion problems [7].…”

Section: (2b)mentioning

confidence: 99%

A Posteriori Modeling Error Estimates for the Assumption of Perfect Incompressibility in the Navier--Stokes Equation

Fischer¹

2015

SIAM J. Numer. Anal.

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We derive a posteriori estimates for the modeling error caused by the assumption of perfect incompressibility in the incompressible Navier-Stokes equation: Real fluids are never perfectly incompressible, but always feature at least some low amount of compressibility. Thus, their behavior is described by the compressible Navier-Stokes equation, the pressure being a steep function of the density. We rigorously estimate the difference between an approximate solution to the incompressible Navier-Stokes equation and any weak solution to the compressible Navier-Stokes equation in the sense of Lions (without assuming any additional regularity of solutions). Heuristics and numerical results suggest that our error estimates are of optimal order in case of "well-behaved" flows and divergence-free approximations of the velocity field. Thus, we expect our estimates to justify the idealization of fluids as perfectly incompressible also in practical situations.

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Section: (2b)mentioning

confidence: 99%

A Posteriori Modeling Error Estimates for the Assumption of Perfect Incompressibility in the Navier--Stokes Equation

Fischer¹

2015

SIAM J. Numer. Anal.

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“…Currently, the conventional thin shell theory (Fluegge, 1973;Love, 2013;Tzou and Howard, 1994), which is based on the Kirchhoff-Love hypothesis ignoring transverse shear deformation, is often used to study shell problems (Confalonieri et al, 2015;Kiendl et al, 2009;Zukas, 1974;Zareh and Qian, 2018;Creaghan and Palazotto, 1994), and for shell structures with small shear stiffness (i.e. prone to significant transverse shear deformation), certain errors will be introduced (Hansbo and Larson, 2011;Repin and Sauter, 2010;Weise, 2018;Rychter, 1988). In addition, the increase in wall thickness of the plate and shell structure often results in exceedance of the application scope of the thin-wall theory in practical engineering, and the influence of transverse shear deformation must be considered.…”

Section: Introductionmentioning

confidence: 99%

“…, 2009; Zukas, 1974; Zareh and Qian, 2018; Creaghan and Palazotto, 1994), and for shell structures with small shear stiffness (i.e. prone to significant transverse shear deformation), certain errors will be introduced (Hansbo and Larson, 2011; Repin and Sauter, 2010; Weise, 2018; Rychter, 1988). In addition, the increase in wall thickness of the plate and shell structure often results in exceedance of the application scope of the thin-wall theory in practical engineering, and the influence of transverse shear deformation must be considered.…”

Section: Introductionmentioning

confidence: 99%

Mesh refinement of finite element method for free vibration analysis of variable geometrical rotating cylindrical shells

Wang

2023

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PurposeThis study aims to provide a reliable and effective algorithm that is suitable for addressing the problems of continuous orders of frequencies and modes under different boundary conditions, circumferential wave numbers and thickness-to-length ratios of moderately thick circular cylindrical shells. The theory of free vibration of rotating cylindrical shells is of utmost importance in fields such as structural engineering, rock engineering and aerospace engineering. The finite element method is commonly used to study the theory of free vibration of rotating cylindrical shells. The proposed adaptive finite element method can achieve a considerably more reliable high-precision solution than the conventional finite element method.Design/methodology/approachOn a given finite element mesh, the solutions of the frequency mode of the moderately thick circular cylindrical shell were obtained using the conventional finite element method. Subsequently, the superconvergent patch recovery displacement method and high-order shape function interpolation techniques were introduced to obtain the superconvergent solution of the mode (displacement), while the superconvergent solution of the frequency was obtained using the Rayleigh quotient computation. Finally, the superconvergent solution of the mode was used to estimate the errors of the finite element solutions in the energy norm, and the mesh was subdivided to generate a new mesh in accordance with the errors.FindingsIn this study, a high-precision and reliable superconvergent patch recovery solution for the vibration modes of variable geometrical rotating cylindrical shells was developed. Compared with conventional finite element method, under the challenging varying geometrical circumferential wave numbers, and thickness–length ratios, the optimised finite element meshes and high-precision solutions satisfying the preset error limits were obtained successfully to solve the frequency and mode of continuous orders of rotating cylindrical shells with multiple boundary conditions such as simple and fixed supports, demonstrating good solution efficiency. The existing problem on the difficulty of adapting a set of meshes to the changes in vibration modes of different orders is finally overcome by applying the adaptive optimisation.Originality/valueThe approach developed in this study can accurately obtain the superconvergent patch recovery solution of the vibration mode of rotating cylindrical shells. It can potentially be extended to fine numerical models and high-precision computations of vibration modes (displacement field) and solid stress (displacement derivative field) for general structural special value problems, which can be extensively applied in the field of engineering computations in the future. Furthermore, the proposed method has the potential for adaptive analyses of shell structures and three-dimensional structures with crack damage. Compared with conventional finite element methods, significant advantages can be achieved by solving the eigenvalues of structures with high precision and stability.

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“…[1,4,5]). Estimates of the different type that can be used to evaluate modeling errors for problems on domains with a given thickness were derived in [13] for diffusion problems, in [12] for stressed planes and in [9] for stationary reaction-diffusion problems.…”

Section: Introductionmentioning

confidence: 99%

Combined Error Estimates in the Case of Dimension Reduction

Samrowski

2014

Computational Methods in Applied Mathematics

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-We consider the stationary reaction-diffusion problem in a domain Ω ⊂ R 3 having the size along one coordinate direction essentially smaller than along the others. By an energy type argumentation, different simplified models of lower dimension can be deduced and solved numerically. For these models, we derive a guaranteed upper bound of the difference between the exact solution of the original problem and a three-dimensional reconstruction generated by the solution of a dimensionally reduced problem. This estimate of the total error is determined as the sum of discretization and modeling errors, which are both explicit and computable. The corresponding discretization errors are estimated by a posteriori estimates of the functional type. Modeling error majorants are also explicitly evaluated. Hence, a numerical strategy based on the balancing modeling and discretization errors can be derived in order to provide an economical way of getting an approximate solution with an a priori given accuracy. Numerical tests are presented and discussed.2010 Mathematical subject classification: 35J20, 65N15, 65N30.

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Estimates of the modeling error for the Kirchhoff–Love plate model

Cited by 4 publications

References 12 publications

A Posteriori Modeling Error Estimates for the Assumption of Perfect Incompressibility in the Navier--Stokes Equation

A Posteriori Modeling Error Estimates for the Assumption of Perfect Incompressibility in the Navier--Stokes Equation

Mesh refinement of finite element method for free vibration analysis of variable geometrical rotating cylindrical shells

Combined Error Estimates in the Case of Dimension Reduction

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