Plenary
Lecture
Accurate Element Method Methodology for Finding and
Controlling the QuasiAnalytic Solutions of FirstOrder
Partial Differential Equations with Variable
Coefficients
Professor Maty Blumenfeld
Politehnica University Bucharest
ROMANIA
Abstract:
A first order Partial Differential Equation (PDE) with
variable coefficients has to be integrated on a domain
D, divided in rectangular subdomains (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 NorthEast
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 quasianalytic
solution represented by the CF, introduces a
fundamentally different strategy that can be summarized
as “best shapecontrolled 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 quasianalytic 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 ?1011 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,
19471952
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
nonofficially in 1980 and officially in 1990)
b. Guidance for many Ph.D. works (Strength of materials,
theory of elasticity and plasticity)
c. President (19901995) 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)
