Plenary Lecture

Plenary Lecture

Accurate Element Method Methodology for Finding and Controlling the Quasi-Analytic Solutions of First-Order Partial Differential Equations with Variable Coefficients

Professor Maty Blumenfeld
Politehnica University Bucharest

Abstract: A first order Partial Differential Equation (PDE) with variable coefficients has to be integrated on a domain D, divided in rectangular sub-domains (elements). The four sides of the element – having the dimensions B(base) ? H(height) – will be referred as South, West, North and East edges, respectively. A known initial condition is considered on the South Edge and a boundary condition on the West Edge. Integrated symbolically the PDE leads to an integral equation. The Accurate Element Method (AEM) performs the integration replacing the unknown solution by a Concordant Function (CF), which is a complete two variables polynomial of high degree. The number of terms of a complete G degree CF is given by NT=(G+1) (G+2)/2. For instance a five degree CF includes 21 terms. In order to obtain RIGOROUSLY 21 equations AEM uses three sources:
1. The integral equation (1 equation)
2. The initial and boundary conditions (11 equations)
3. The PDE itself and its derivatives applied in the nodes of the element (9 equations). Because the North-East node – referred as Target Node – is included in these equations, the AEM is an implicit method, unconditionally stable and allowing the use of elements with large dimensions that can be considered as improper by other methods.
The standard strategy for the numerical integration of PDEs is usually based on two considerations: “the shape of the element has to be close to a square” and b. “the precision improves when the number of elements increases”. The AEM that obtains for each element a quasi-analytic solution represented by the CF, introduces a fundamentally different strategy that can be summarized as “best shape-controlled accuracy“:
A. Best shape. Many examples solved by using CFs have shown that only in some particular cases the element has to be square. The AEM introduces a more strong analysis from which it results that the shape of the element has to be adapted to the particular case of the PDE to be integrated. This analysis is based on the facility offered by the quasi-analytic solution, which can be replaced in the PDE leading to residual functions evaluated as root mean square values on the North and East sides of the element. Based on the ratio of these residuals one can establish the best shape of the element for each particular. The correctness of this evaluation can be linked to the characteristic curves of the PDE. For instance, for the particular case of a PDE with variable coefficients the standard strategy based on square elements lead – when the number of elements were increased – to erratic results, while the AEM strategy based on rectangular elements having the ratio H/B=7 lead to strictly convergent results.
B. Accuracy check. The AEM can closely check the accuracy of the Target Value in two ways: based on the root mean square values of the residual functions and by successively computing the Target Value with two or many Concordant Functions. For a particular example for which the root mean square value was RMS ?10-11 it resulted:
Five degree CF with 21 terms: Target value = 5.453688724546589
Seven degree CF with 36 terms: Target value = 5.453688723145416
The two values coincide with 9 digits, so that one can consider 5.45368872 as reliable.

Brief Biography of the Speaker:
1. Born: 15 august 1928
2. Education: Engineer, Politehnica, Bucharest, 1947-1952
3. Ph.D., 1964:”The three unknowns method applied to the mechanic systems”. Further studies of the method developed also by other authors. The method has been presented in different books and is used for the “Strength of materials” lectures.
4. Activities:
a. Professor since 1970 at the “Strength on Materials” department
- Strength of materials
- Finite Element Method (Has introduced for the first time in Politechnica University Bucharest the FEM course non-officially in 1980 and officially in 1990)
b. Guidance for many Ph.D. works (Strength of materials, theory of elasticity and plasticity)
c. President (1990-1995) of SIAC (Society for the computer aided engineering). Honorary President since 1995.
d. Scientific counselor INAS (Institute for the System Analysis) – Craiova (Romania)
e. More than 100 research and experimental works for industry and research institute.
f. A great number of papers, studies and 12 books (from which 6 as single author)


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