The solution of 3D models in degenerated geometries in which some characteristic dimensions are much lower than the other ones -e.g. beams, plates, shells,...- is a tricky issue when using standard mesh-based discretization techniques. Separated representations allow decoupling the meshes used for approximating the solution along each coordinate. Thus, in plate or shell geometries 3D solutions can be obtained from a sequence of 2D and 1D problems allowing fine and accurate representation of the solution evolution along the thickness coordinate while keeping the computational complexity characteristic of 2D simulations. In a former work this technique was considered for addressing the 3D solution of thermoelastic problems defined in plate geometries. In this work, the technique is extended for addressing the solution of 3D elastic problems defined in shell geometries. The capabilities of the proposed approach are illustrated by considering some numerical examples involving different degrees of complexity, from simple shells to composite laminates involving stiffeners. The analyzed examples prove the potentiality and efficiency of the proposed strategy, where the computational complexity was found evolving as reported in our former works, proving that 3D solutions can be computed at a 2D cost.

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Separated representations of 3D elastic solutions in shell geometries

http://web.mit.edu/kjb/www/Principal_Publications/3D_shell_elements_for_structures_in_large_strains.pdf

3D-shell elements for structures in large strains

This paper is concerned with the development of a new family of solid–shell finite elements. This concept of solid–shell elements is shown to have a number of attractive computational properties as compared to conventional three-dimensional elements. More specifically, two new solid–shell elements are formulated in this work (a fifteen-node and a twenty-node element) on the basis of a purely three-dimensional approach. The performance of these elements is shown through the analysis of various structural problems. Note that one of their main advantages is to allow complex structural shapes to be simulated without classical problems of connecting zones meshed with different element types. These solid–shell elements have a special direction denoted as the “thickness”, along which a set of integration points are located. Reduced integration is also used to prevent some locking phenomena and to increase computational efficiency. Focus will be placed here on linear benchmark problems, where it is shown that these solid–shell elements perform much better than their counterparts, conventional solid elements.

https://hal.archives-ouvertes.fr/hal-01206680/document

New quadratic solid–shell elements and their evaluation on linear benchmark problems

This paper presents the development of a new three-dimensional brick finite element by the use of the strain based approach for the linear analysis of plate bending. The developed element has the three essential external degrees of freedom (U, V and W) at each of the eight corner nodes as well as at the centroidal node. The displacement field of the developed element is based on assumed functions for the various strains satisfying the compatibility equations and the static condensation technique is used for the internal node. The performance of this element is evaluated on several problems related to thick and thin plate bending in linear analysis. The obtained results show the good performances and accuracy of the present element.

A new strain based brick element for plate bending

In this article, hexahedral piezoelectric solid–shell finite element formulations with linear and quadratic interpolation, denoted by SHB8PSE and SHB20E, respectively, are proposed for the modeling of piezoelectric sandwich structures. Compared to conventional solid and shell elements, the solid–shell concept reveals to be very attractive, due to a number of well-established advantages and computational capabilities. More specifically, the present study is devoted to the modeling and analysis of multilayer structures that incorporate piezoelectric materials in the form of layers or patches. The interest in this solid–shell approach is shown through a set of selective and representative benchmark problems. These include numerical tests applied to various configurations of beam, plate and shell structures, both in static and vibration analysis. The results yielded by the proposed formulations are compared with those given by state-of-the-art piezoelectric elements available in ABAQUS, in particular,...

/pdf/linear-and-quadratic-solid-shell-finite-elements-shb8pse-and-13jju9qq9q.pdf

Linear and quadratic solid–shell finite elements SHB8PSE and SHB20E for the modeling of piezoelectric sandwich structures

This paper presents the development of a new prismatic solid-shell finite element, denoted SHB6, obtained using a purely three-dimensional approach. This element has six nodes with displacements as the only degrees of freedom, and only requires two integration points distributed along a preferential direction, designated as the “thickness”. Although geometrically three-dimensional, this element can be conveniently used to model thin structures while taking into account the various phenomena occurring across the thickness. A reduced integration scheme and specific projections of the strains are introduced, based on the assumed-strain method, in order to improve performance and to eliminate most locking effects. It is first shown that the adopted in-plane reduced integration does not generate “hourglass” modes, but the resulting SHB6 element exhibits some shear and thickness-type locking. This is common in linear triangular elements, in which the strain is constant. The paper details the formulation of this element and illustrates its capabilities through a set of various benchmark problems commonly used in the literature. In particular, it is shown that this new element plays a useful role as a complement to the SHB8PS hexahedral element, which enables one to mesh arbitrary geometries. Examples using both SHB6 and SHB8PS elements demonstrate the advantage of mixing these two solid-shell elements.

/pdf/a-new-assumed-strain-solid-shell-formulation-shb6-for-the-2dgmk95gf4.pdf

A new assumed strain solid-shell formulation “SHB6” for the six-node prismatic finite element

Because accuracy and efficiency are the main features expected within the finite element (FE) method, the current contribution proposes a six-node prismatic solid–shell, denoted (SHB6). The formulation is extended here to geometric and material nonlinearities, and focus will be placed on its validation on nonlinear benchmark problems. This type of FE is specifically designed for the modeling of thin structures, by combining several useful shell features with some well-known solid element advantages. Therefore, the resulting derivation only involves displacement degrees of freedom as it is based on a fully 3D approach. Some of the motivation behind this formulation is to allow a natural mesh connection in problems where both structural (shell/plate) and continuum (solid) elements need to be simultaneously used. Another major interest of this prismatic solid–shell is to complement meshes that use hexahedral solid–shell FE, especially when free mesh generation tools are employed. To achieve an efficient formulation, the assumed-strain method is combined with an in-plane one-point quadrature scheme. These techniques are intended to reduce both locking phenomena and computational cost. A careful analysis of possible stiffness matrix rank deficiencies demonstrates that this reduced integration procedure does not induce hourglass modes and thus no stabilization is required.

/pdf/assumed-strain-solid-shell-formulation-for-the-six-node-29h0gt1gta.pdf

Assumed-strain solid-shell formulation for the six-node finite element SHB6: Evaluation on non-linear benchmark problems

This paper presents the development of a six-node solid-shell finite element called (SHB6) and based on the assumed strain method adopted by Belytschko et al. [2]. It is integrated with a set of five Gauss points along a special direction, denoted “thickness”, and with only one point in the other in-plane directions. Its discrete gradient is modified in order to attenuate shear and membrane locking. A series of popular linear benchmark problems has been carried out with comparisons to geometrically similar, low-order three-dimensional elements.

https://hal.archives-ouvertes.fr/hal-01224492/document

New prismatic solid-shell element: Assumed strain formulation and evaluation on benchmark problems

Over the last decade, considerable progress has been made in the development of three-dimensional finite elements capable of modeling thin structures. The coupling between solid and shell formulations has proven to be an interesting way to provide continuum finite element models that can be efficiently used for structural applications. The current work proposes the formulation of two solid-shell elements based on a purely three-dimensional approach. These elements have numerous advantages for the analysis of various complex structural geometries that are common in many industrial applications. Their main advantage is to allow such complex structural shapes to be meshed without classical problems of connecting zones meshed with different element types (continuum and structural elements for instance). Another important benefit of solid-shell elements is the avoidance of tedious pure-shell element formulations needed for the complex treatment of large rotations. The two solid-shell elements developed are a 20-node and a 15-node element, respectively, with displacements as the only degrees of freedom. They also have a special direction called “the thickness”. Therefore, they can be used for the modeling of thin structures, while providing an accurate description of various through-thickness phenomena thanks to the use of a set of integration points in that direction. A reduced integration scheme has been introduced to prevent some locking phenomena and increase computational efficiency. To assess the effectiveness of the proposed solid-shell elements, a set of popular benchmark problems is investigated, involving linear as well as geometric nonlinear analyses. It is shown that these elements can support high aspect ratios, up to 500, and are especially efficient for elastoplastic bending behavior. The various numerical experiments in linear and nonlinear situations reveal that these solid-shell elements perform really better than standard solid elements having similar properties in terms of geometry, interpolation and degrees of freedom.

Vuong-Dieu Trinh

Papers

A new assumed strain solid-shell formulation “SHB6” for the six-node prismatic finite element

New quadratic solid–shell elements and their evaluation on linear benchmark problems

Assumed-strain solid-shell formulation for the six-node finite element SHB6: Evaluation on non-linear benchmark problems

New prismatic solid-shell element: Assumed strain formulation and evaluation on benchmark problems

Formulation of new quadratic solid-shell elements and their evaluation on popular benchmark problems