New Unsymmetric Finite Element Method On Plane And Plate Problems

The finite element method(FEM)has been widely used in numerical simulation and structural design of various engineering problems.With the development of computers and more and more functions provided by commercial software,FEM has been one of the dominant numerical methods in most engineering fields.Nevertheless,the traditional FEM has some shortcomings.For example,the isoparametric finite elements based on the principle of minimum potential energy are extremely sensitive to the mesh distortions.Once the geometry shapes of elements are irregular,the accuracy of numerical results will be significantly reduced.Besides,the existence of incompressible materials,thin plates or other problems often results in the numerical locking.In response to these problems,many scholars devote themselves to it and put forward various FEM theories and element models,which make great progress in high-performance FEM.However,there are still more or less limitations in existing element models,which means the problems haven't been solved from the bottom.Recent studies show that the unsymmetric FEM performs well even if the discretized meshes are severely distorted,but the original unsymmetric FEM is not perfect.In this dissertation,a novel unsymmetric FEM is developed and successfully applied to plane and plate problems.The new element model has excellent performance which is resistant to grid distortions and avoids volumetric locking and shear locking.The main work is as follows:First,a novel incompatible and unsymmetric four-node quadrilateral plane element IUQ4 is presented using both incompatible trial functions and test functions based on the framework of unsymmetric FEM.Five internal nodes,one at the elemental central and four at the middle sides,are added to ensure the quadratic completeness of the elemental displacement field.In addition,a local skew frame is attached to the element center to avoid the rotational variance.Under the skew frame,the trial shape functions with quadratic completeness can be obtained.And the test shape functions consist of the standard isoparametric interpolation functions with additional incompatible shape functions.The incompatible test shape functions are directly taken from the 9-node Lagrange interpolation functions and modified to pass the patch test.Therefore,the unsymmetric elemental stiffness matrix can be obtained with two sets of different interpolation functions.Numerical results show that the present element not only retains the advantage of good immunity to mesh distortion from the original unsymmetric finite elements,but also possesses the rotational frameinvariance and avoids the trapezoidal locking and volumetric locking.Second,the new unsymmetric FEM is extended to the Reissner-Mindlin plate.Adding an internal node and a local skew frame at the element center,a new unsymmetric eight-node quadrilateral plate element IUQ8 is finally constructed.Numerical results show that the present element not only keeps the merit of insensitivity to mesh distortion and possesses rotational frame invariance,but also free of shear locking when the plates are very thin.It also has no difficulty when the element shapes degenerate into triangles.Finally,the new unsymmetric FEM by introducing internal displacement relaxes the continuity requirements of the virtual displacements between element interfaces and avoids the difficulty that low-order elements cannot guarantee high-order completeness due to the lack of degrees of freedom.Unlike the existing unsymmetric elements with introduction of analytical stress solutions,the present method is more convenient to extend to the analysis of axisymmetric and 3D solid problems.