engleski

Review of Adaptive Fup Collocation Method

In the past 15 years, we were investigating procedures of applications of the atomic functions in computational modeling of different engineering problems. This review presents some applications of the Fup basis functions in numerical analyses of structural problems. Numerical models were created by applying the collocation method (Fup Collocation Method - FCM). Because of the property of universality, it is possible to hierarchically increase the number of Fup basis functions during calculation (multilevel approach). We developed hierarchic generation of the solutions of non-linear problems first in 1D problem of elasto-plastic analyses of a beam bending, then on 2D examples of elasto-plastic behavior of a prismatic bar subjected to torsion. The results obtained by the FCM are compared with the existing exact solutions and numerical solutions obtained by the FEM. Results of our investigations show some advantages of the proposed method: predefined mesh and numerical integration are avoided ; values of the main solution function and all values derived from the main solution are calculated in the same points with the same level of accuracy ; all fields derived from the main solution can be expressed by continuous functions on the entire domain ; the real non linear behavior is efficiently simulated: the multilevel approach provides a simple way to increase the accuracy of an approximate solution in places where plastic yielding occurs and also accelerates the convergence of incremental-iterative procedure. Improvement of the Fup Collocation Method was continued by developing of a multi-resolution adaptive algorithm for solving problems described by partial differential equations. The distribution of collocation points within the observed area is changed adaptively, depending on the character of solution function and accuracy criteria. The space-time Adaptive Fup Collocation Method (AFCM) for solving boundary-initial value problems is presented. To solve the one-dimensional initial boundary value problem, we convert the problem into a two-dimensional boundary value problem. This quasi-boundary value problem is then solved simultaneously in the space-time domain with a collocation technique and by using atomic Fup basis functions. The method is very convenient for solving engineering problems with sharp fronts and narrow transition zones that are changed in space and time.

atomic basis functions; universality; collocation method; multilevel base; adaptive technique

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