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Elements: Element Formulation
Three types of triangular surface elements, viz the CT, LT and ST elements shown in the figure below are available in Examine3D for the discretization of excavation surfaces. The initial or undeformed shape of all three types of elements is planar; however, the variation of the surface variables, viz the displacements and tractions, is assumed to be constant, linear and quadratic over the CT, LT and ST elements, respectively. Thus, the CT element is superparametric, the LT element is isoparametric and the ST element is subparametric. With initially planar shapes, the elements available in the program can model flat excavation surfaces exactly. Curved surfaces, which generally exist in civil engineering structures, can be fairly approximated with these elements if a sufficiently fine mesh is used.
Element library of Examine3D
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Elements: Closed-form Integration Close to Boundary
In Examine3D, singular surface integrals are evaluated for the CT, LT and ST elements using a combination of the row-sum elimination method and the regularizing transformation method. The non-singular integrals, i.e. the nearly singular and the regular integrals, are computed with a new hybrid integration scheme developed in this software. This scheme uses a combination of analytic integration and Gauss-Hammer numerical integration of different orders to attain accuracy and economy in evaluating these integrals. For any load point-surface element pair, the scheme chooses the type or order of integration on the basis of a rational selection criterion that takes into account the distance between the load point and the element as well as the element's size and shape.
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Elements: Conforming Mesh
A conforming mesh requires that every element edge be shared by only one other element. The formulation of the mesh also ensures a continuity of displacement along the element edge. The degrees of freedom are located at the vertices or nodes of the element, and each node is shared by all elements attached to it. The advantage in using a triangular conforming mesh is related to the nodal sharing formulation. The reduction in nodes reduces the degrees of freedom, thereby decreases the computation time and amount of memory required to store the matrix. In addition, our software developers demonstrated that one could expect a more accurate result using conforming meshes compared to nonconforming meshes for a given equivalent degrees of freedom. A triangular element is used because it has a simpler formulation with an ability, as a simple geometric primitive, to more easily model almost any three-dimensional surface. Although the complexity of modeling three-dimensional objects is greatly increased by the use of a conforming mesh (as oppose to nonconforming mesh), the computational advantages of the former outweigh any disadvantages.
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Elements: Virtual Number of Element Free Surfaces
A host of direct and iterative solvers are available in Examine3D for the solution of the system of the linear algebraic equations resulting from the discretization of the boundary integral equations. These equation solvers are based on the LU decomposition method, the relaxed Jacobi method, the conjugate gradient method, the Jacobi / bi-conjugate gradient method and the Generalized Minimum Residual method. Apart from the LU decomposition solver, all the others are iterative solvers and are capable of performing out-of-core solution.
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