Two higher order hybrid-Trefftz elements for thin plate bending analysis

Summary The hybrid‐Trefftz stress element is used to emulate conventional finite elements for analysis of Kirchhoff and Mindlin‐Reissner plate bending problems. The element is hybrid because it is based on the independent approximation of the stress‐resultant and boundary displacement fields. The Trefftz variant is consequent on the use of the formal solutions of the governing Lagrange equation to approximate the stress‐resultant field. In order to emulate conventional elements, nodal functions are used to approximate the displacements on the boundary of the element. Duality is used to set up the element solving system. The associated variational statements and conditions for existence and uniqueness of solutions are recovered. Triangular and quadrilateral elements are tested and characterized in terms of convergence, sensitivity to shear‐locking, and shape distortion. Their relative performance is assessed using assumed strain Mixed Interpolation of Tensorial Components (MITC) elements and recently proposed Trefftz‐based elements. This relative assessment is extended to a hypersingular problem to illustrate the effect of enriching the domain and boundary approximation bases.

Section: Numerical Testsmentioning

confidence: 96%

Section: Numerical Implementationmentioning

confidence: 99%

{bold-italicZ}_{normals} = {bold-italicZ}_{normalEB} = [\begin{matrix} center \\ \begin{array}{c} Z_{- 2} Z_{1}^{2} & 0 & - {LsZ}_{1}^{2} \\ 0 & Z_{1} & 0 \\ ((), 6 false/ L) {sZ}_{1} & 0 & Z_{1} Z_{3} \end{array} & \begin{array}{c} 3 s^{2} Z_{\frac{2}{3}} & 0 & 3 s^{2} Z_{\frac{2}{3}} \\ 0 & s & 0 \\ - ((), 6 false/ L) {sZ}_{1} & 0 & - 2 {sZ}_{\frac{3}{2}} \end{array} \end{matrix}],

{bold-italicZ}_{normals} = {bold-italicZ}_{normalEB} + {italicsZ}_{1} \frac{γ}{1 + γ} [\begin{matrix} center \\ \begin{array}{c} - Z_{2} & 0 & \frac{1}{2} {LZ}_{2} \\ 0 & 0 & 0 \\ - 6 false/ L & 0 & 3 \end{array} & \begin{array}{c} Z_{2} & 0 & \frac{1}{2} {LZ}_{2} \\ 0 & 0 & 0 \\ 6 false/ L & 0 & 3 \end{array} \end{matrix}] .

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Section: Numerical Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The first application to thick plates is due to Petrolito . Applications to isotropic and anisotropic problems are reported in References .…”

Section: Introductionmentioning

confidence: 99%

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2020

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Bussamra

Neto

2017

Summary Three‐dimensional hybrid‐Trefftz stress finite elements for plates and shells are proposed. Two independent fields are approximated: stresses within the element and displacement on their boundary. The required stress field derived from the Papkovitch‐Neuber solution of the Navier equation, which a priori satisfies the Trefftz constraint, is generated using homogeneous harmonic polynomials. Restriction on the polynomial degree in the coordinate measured along the thickness direction is imposed to reduce the number of independent terms. Explicit expressions of the corresponding independent polynomials are listed up to the fifth order. Illustrative applications to evaluate displacements and stresses are conducted by hexahedral hybrid‐Trefftz stress element models. The hierarchical p‐ and h‐refinement strategy are exploited in the numerical tests.

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2019

A novel smart hybrid-Trefftz finite element (HTFE) has been developed for the analysis of smart laminated composite plates. The substrates of the smart plates are symmetric and antisymmetric cross-ply plates. The derivation of this HTFE is devoid of the complicated task of finding the particular solutions of simultaneous governing partial differential equations. The Trefftz functions are constructed from the finite number of free-field exact solutions of the homogeneous simultaneous governing partial differential equations of the element domain in a straightforward manner without transforming them into a single governing equation. The HTFE is validated with the exact solutions of the smart composite plates. It is observed that this HTFE is an efficient finite element and can be utilized for the analysis of active control of smart composite structures. KEYWORDS composites, finite element methods, smart materialsInt J Numer Methods Eng. 2019;120:707-726.wileyonlinelibrary.com/journal/nme