Theory Investigation Of Novel Low Order Elements With High Accuracy For Mechanical Structure Analysis

The finite element method meets low accuracy or can not work for some engineering practice problems, such as metal forming、vehicle body panel forming and crashing, containing dynamic large deformation analysis. Meshless methods can work perfectly for large deformation analysis, but they encounter the major technical barrier-low computation efficiency. To solve these shortages of existing numerical method, this thesis propose a novel numerical theory system, which are very effective for engineering analysis. To pursue this goal, this thesis present several effective numerical method and plate and shell elements based on weakend weak form and generalized gradient smoothing technique. Some important properties, such as accuracy and convergence, are proved, and a numerical theory system is constructed. Main work includes the following aspects:1. Novel numerical methods based on weakend weak formThe subdomain gradient smoothing technique and a cell-based smoothing radial point interpolation method are proposed in this thesis. Conventional gauss integration becomes line integration along smoothing subdomain edges. The supporting node selection for shape function construction is based on the background cells, which incorporating the meshless interpotation can deal with the mesh distortion effectively, and also has a high accuracy and efficiency.An edge-based smoothed finite element method (ES-FEM) for elastic-plastic analysis is formulated. Theorem-proof of the results demonstrate that the edge-based smoothing operation reduces the stiffness of the discretized system, and compensates nicely the "over-stiffness" of the FEM model. The effective elastic modulus and Possion’s ratio in each smoothing domain are obtained for computing the elastic-plastic problems, and the effective material parameters are the real material parameters of the hardening material.2. Proposed several novel Mindin plate and shell elementsA formulation of Mindlin-Reissner plates is proposed using the cell-based smoothed radial point interpolation method with sub-domain smoothing operations. Effective treatment for shear-locking are given and a plate formulation with high accuracy of stress is obtained.A novel quadrilateral plate and shell element is presented based on the smoothed finite element method. An average shape function is proposed, and shear locking phenomenon is avoided by using only one smoothing cell. Area integration over each smoothing cells is recast into line integration along its edges, and the shape functions are obtained directly in the Cartesian coordinate, hence no mapping is needed and extremely distorted elements can be used. This quadrilateral plate and shell element provides very stable and accurate results with the low computational effort compared, which is very effective for engineering analysis.A novel edge-based smoothed triangular plate and shell element theory is proposed, and an edge local coordinate system is introduced for performing strain smoothing operations for the elements consisting the smoothing domain not in a plane. The present method can be easily implemented with little changes to the esxisting triangular shell elements. The new element is very high accuracy and not sensitive to mesh distortion, which is very useful to engineering analysis.3. Novel thin plate and shell element theory using linear interpolationThis thesis presents a way to solve the4th order boundary value problems using simple linear point interpolation method, and a rotation-free Euler-Bernoulli beam element is proposed. A novel numerical theory is formed, and the feasibility of this theory is proved.A novel thin plate element theory is proposed based on a continuity re-relaxed technique to overcome the high continuity requirement for thin plate formulation which need the high computation cost. The curvatures over integration domains are restructured through the divergence theorem, and the continuity requirement of the trial deflection function for thin plate problems can be re-relaxed. Using the proposed continuity re-relaxed technique, the weak form of a2k order PDE can be modeled using shape functions of k-1consistence. The C1continuity thin plate problem can be easily formulated and stable results can be obtained only using Co interpolating functions.Based on the radial point interpolation method and triangular background cells, several types of smoothing domains are constructed, and different thin plate formulation using continuity re-relaxed technique. The properties of the different smoothing domains constructing scheme are discussed, the efficiency and accuracy of the novel thin plate element theory are confirmed.A curvature-constructed method (CCM) for thin plate problems is proposed using three-node triangular cells and assumed piecewisely linear deflection field. The slopes at special points are obtained using the gradient smoothing techniques (GST) over different smoothing domains. The curvature in each cell can be constructed using these slopes, and then used to create the discretized system equations. Some schemes are devised using the present CCM and the numerical results are presented, to test the efficiency and accuracy of these schemes.Outstanding scheme is choosed for constructing shell formulation.4. Dynamic explicit formulation of proposed numerical method and shell element for large deformation analysisAn edge-based gradient smoothing based on weakened weak form is proposed for the simulation of metal forming processes, which involves geometric, material, and contact non-linear. A simple but general contact searching algorithm is used to treat the contact interface and an algorithm for the contact force is presented. The smoothed deformation gradient and the smoothed stress tensor are derived at the reference (or initial) configuration. The integration domains are different from the interpolation domain, and the method can work for extremely distorted meshes. Solutions to upsetting, extrusion of metals with large material distortions are given to show the effectiveness and efficiency of the proposed method.The explicit dynamic formulations of two outstanding triangular shell elements are extended for large deformation analysis. The nodel internal force, external force and inertial force are obtained using the weakened-weak form. Various numerical tests demonstrate the accuracy and efficiency of the proposed triangular shell elements.