Gauss points

GAUSSP. Gives Gauss point coordinates and weights required in the numerical integration of the members of the elemental stiffness equations. 

The quadrature is exact when/is a polynomial of degree 2m — 1 in x. Because there are m weights andm Gauss points, we nave 2m parameters that are chosen to exac tly represent a polynomial of degree 2m — 1, which has 2m parameters. The Gauss points and weights are given in the table. 

Had we used a higher precision for the Gauss points, the quadrature would have rendered a solution even closer to the exact integral. 

Determine the velocity profile and traction profiles in a pressure driven slit flow of a Newtonian fluid. Use Ap =1000 Pa, //, =1000 Pa-s, h 1 mm and a distance from entrance to exit of 1000 mm. Solve the problem using isoparametric 2D quadratic elements and different gauss points, compare your solutions with the analytical solution for slit flow. 

The integrals are evaluated by numerical quadrature, with the local mean curvature H prescribed at each Gauss point. The equations can be solved for the j s by Newton iteration. 

If H is specified as a function H(u, v) of the surface coordinates, then this function is evaluated at each Gauss point where the value H(P ) is needed in those residual equations Rj satisfying (j>j(P 0. Clearly a more rapidly-varying function will require a finer grid. If H is a specified function of spatial position H(r), then this is rewritten, via Eqs. (9) and (10) as a function of u, v, and the solution w (u,v). Examples of this are given in... 

Section IV. In principle, it is possible to prescribe H as a function of position and slope by including in the prescribed function the values of dw /du and dw ldv already calculated at the Gauss points, though this has not been tried. When H is prescribed as a function of position and/or slope, then the Jacobian matrix should include the dependence of this prescribed H on the nodal values however, if this function is slowly-varying and the user is willing to sacrifice quadratic convergence, this contribution to the Jacobian can be neglected or approximated. 

In practice it was found that a numerical Gauss-Legen-dre integration with four Gauss-points [104.] was equivalent or better as soon as... 

We observed that the required number of Gauss points to obtain the same results as with 64. Gauss points, was nearly independent of the length of the element. Therefore all results could be put together in one (worst-case) table (table 3.1), in which the minimum number of Gauss points can be found as function of the (reduced) coordinates (r, z ) of that nodal point of an element closest to the field point with reduced coordinates (1.,0.). For practical use, this table should still be simplified. 

V/hen b is large, soon numerical round-off errors are involved. Therefore a combination of analytic and numerical integration is necessary. This numerical integration needs only a few Gauss points. 

It should be noted that the dissipating (CeUe) and elastic (Keiie) part of the internal forces are calculated by evaluating form function derivatives, material parameters and statefields in the Gauss points, while the inertial contribution is computed through a volume scalar product. 

The distinct feature of elastic-plastic finite element computations is the presence of two iteration levels. In a standard displacement based finite element implementation, constitutive driver at each integration (Gauss) point iterates in stress and internal variable space, computes the updated stress state, constitutive stiffness tensor and delivers them to the finite element functions. Finite element functions then use the updated stresses and stiffness tensors to integrate new (internal) nodal forces and element stiffness matrix. Then, on global level, nonlinear equations are iterated on until equilibrium between internal and external forces is satisfied within some tolerance. 

In the case of elastic computations, constitutive driver has a simple task of computing increment in stresses (Aa,y) for a given deformation increment (Ae /), through a closed form equation (Aa,y = E yi/ Ae /). It is important to note that in this case the amount of work per Gauss point is known in advance and is the same for every integration point. If we assume the same number of integration points per element, it follows that the amount of computational work is the same for each element and it is known in advance. 

Here the functions N, are only calculated at representative points (the so-called Gauss points ) by using the weight factors Wp. Position and typical number of the points Xp as well as the according weight factors can be found in the literature, e.g. (Wriggers, 2000). 

The element sizes within the composite adherends were chosen such that two of the integration points of the second layer of elements adjacent to the adhesive were coincident with the interface between the first two plies of the laminate. Therefore, the stresses would be accurately predicted at the point where interlaminar failure was predicted by Adams et al. [17]. These Gauss point values were chosen as they are the only points within the finite element model at which the solution is fully integrated, and are therefore the most accurate predictions of stress and strain. The stresses and strains were monitored at these points in order to predict the interlaminar stresses so that the influence of temperature could be quantified. 

Figure 3 The Linear Elastic Solution o Stresses at Gauss Points Average Stress at Centers...

The normalized bending stresses at the Gauss points and centers of eight elements nearest to the fixed end are plotted as functions of the normalized z-coordinate in Fig.(3-7). The solid line represents the solution of ihe elementary beam theory, which serves as a reference for comparison. It is seen in Fig.3 that the linear elastic solution, which is obtained by multiplying the solution at F=0.01 by one hundred, matches the... 

Figure 4 The Large Strain Elastic Solution o - Stresses at Gauss Points - Average Stresses at Centers...

The magnitude of the stress at the Gauss point, which is nearest to the origin, is shown as a function of the deflection of the beam at the free end, [-Mj(L,0,0)], in Fig. 8 for the entire loading and unloading processes. The differences between Case 1 and Case 2 can be observed. The contrast between the stiffness in the elastic range and that in the plastic range is indicated also. 

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