A Unified Approach to the Finite Element Method and Error by Julian A. T. Dow

By Julian A. T. Dow

This e-book presents an in-depth heritage to higher realizing of finite aspect effects and strategies for bettering accuracy of finite point equipment. therefore, the reader is ready to establish and get rid of error contained in finite aspect types. 3 varied errors research strategies are systematically constructed from a typical theoretical origin: 1) modeling erros in person parts; 2) discretization error within the total version; three) point-wise error within the ultimate rigidity or pressure results.Thoroughly classification confirmed with undergraduate and graduate scholars. A Unified method of the Finite point technique and blunder research systems is certain to turn into an important source for college students in addition to practising engineers and researchers. * New, easier point formula recommendations, model-independent effects, and blunder measures* New polynomial-based tools for picking out severe issues* New strategies for comparing sheer/strain accuracy* obtainable to undergraduates, insightful to researchers, and worthwhile to practitioners* Taylor sequence (polynomial) dependent* Intuitive elemental and point-wise mistakes measures* crucial history info supplied in 12 appendices

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A solution found by minimizing the functional from which the governing differential equations are derived is called a variational solution of that equation. A solution found by directly solving the differential equation is called a classical or analytic solution. The finite element method minimizes an approximation of the functional, therefore, the finite element method is a variational technique. In contrast, the finite difference method approximates the differential equation and then solves the approximation, so it is a classical approach.

Ix Figure 4. Longitudinal bar example. x Figure 5. Variation for the bar problem. Lesson 2 9 29 The top of the bar is fixed so the displacement u(0) is specified as zero. Let us show that this boundary condition satisfies the boundary term at x = 0. Since the displacement is specified and the variation must satisfy the boundary conditions, r/must equal zero at x - 0. Thus, the boundary term at x = 0 given in Eq. 22 is satisfied because one of its components, the q term, is zero. As a result of this argument, geometric or essential boundary conditions satisfy the requirement that the boundary term be zero if the functional is to be minimized.

A virtual displacement is denoted here as 5r. The characteristics of a virtual displacement are explicated by deriving the principle of virtual work for discrete systems and by using a virtual displacement in a problem. The derivation identifies the theoretical reason for the constraints on a virtual displacement. The example problem clearly demonstrates the meaning of the constraints in practice. 1. Admissible virtual displacements. 2. Inadmissible virtual displacements. 2 The idea that a virtual displacement is imaginary does not have any significance in statics applications.

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