FlexPDE
FlexPDE is a unique, flexible and powerful general purpose software system
for obtaining numerical solutions to the coupled sets of partial differential
equations frequently found in engineering, physics, chemistry, biology,
geology, mathematics and other scientific fields.
FlexPDE uses the powerful finite element method to obtain its numerical
solutions. FlexPDE does not, however, require its users to have any knowledge
of the intricacies of the finite element method itself or the mesh generation
associated with it. Users who can express their problem as an initial
value/boundary value problem can use FlexPDE immediately.
FlexPDE is a software tool for the solution of systems of partial differential
equations. It offers an integrated solution environment, including problem
description language, numerical modelling, and graphical output of solutions.
Features
Equation Systems
FlexPDE can treat boundary value and eigenvalue problems in two or three space
dimensions, as well as initial/boundary value problems in two or three space
dimensions plus time. The equations are assumed to be of first or second order
in space, and first order in time. Equations of higher order must be rewritten
as systems of equations of lower order. Equations may be linear or nonlinear,
and FlexPDE will automatically apply a solution method which is appropriate to
the system. The number of simultaneous equations is limited only by the
resources of the computer.
Boundary Conditions
Boundary conditions may be specified as arbitrary combinations of
"value", "natural" , "Neumann",
"periodic" and "antiperiodic" conditions.
- "Value" boundary conditions specify the value of a given
dependent variable as a function of constants, spatial coordinates, and values
or derivatives of dependent variables.
- "Natural" boundary conditions depend for their meaning on
the way the equations are written, but in the usual case refer to the
specification of a boundary flux. Natural boundary conditions are given as
functions of constants, spatial coordinates, and values or derivatives of
dependent variables. Consider for example the heat equation div(-K*grad(T))=H.
Application of the divergence theorem to the left side reduces it to the
surface integral of (-K*grad(T)), which is the meaning of the natural boundary
condition, ie. the surface flux.
- "Neumann" boundary conditions specify the outward normal
derivative of the variable.
- "Periodic" and "antiperiodic" boundary
conditions force the values of points on one boundary to replicate or negate
values on another boundary, identified by an arbitrary coordinate
transformation expression. Linear extension, azimuthal rotation and other shape
periodicities may be specified.
Finite Element Model
FlexPDE uses a Galerkin finite element model, with quadratic or cubic basis
functions involving nodal values of system variables only. This model assumes
that the dependent variables are continuous over the problem domain, but does
not require or impose continuity of derivatives of the dependent variables.
Second-order terms in the equations will give rise to various forms of flux
continuity (through surface integrals generated by integration by parts), and
these conditions will be imposed in an integral sense over the cell faces.
There are ramifications to this model which require care on the part of the
user. In electromagnetics, for example, the normal component of D is continuous
across material discontinuities, while the tangential component of E is
continuous. It is not possible in general to satisfy both of these conditions
if the field components themselves are chosen as the system variables. There
are two ways to address this difficulty. First is to pose the problem in terms
of potentials and not field components. The potential equation
div(eps*grad(V))=rho accurately represents all the physical requirements of the
system. The cell-face integral of the normal component of D will be continuous
across material interfaces (this follows from application of the divergence
theorem to the PDE), and the tangential component of E will be everywhere
continuous on the interface (since V is single-valued on the interface). If the
user is still determined to model field components instead of potentials, then
he must restrict himself to problems in which the continuity reqirements can be
met, ie., in which some of the field components are missing.
Problem Domains
Problem domains can be arbitrarily complex in two space dimensions, but
contiguity is assumed. Two-dimensional domains may be made up of an arbitrary
number of regions, with differing parameter definitions in all or any region.
Three dimensional domains are constructed as layered extrusions of
two-dimensional domains, and so are more restricted. Any number of layers may
be specified, and material parameters may be different in any layer of any
region. Layer interfaces may be non-planar, specified by arbitrary functions of
2D spatial coordinates, but must not intersect.
Adaptive Meshes
FlexPDE automatically generates an unstructured computational mesh of triangles
or tetrahedra which fill the domain and match region boundaries. The initial
mesh is adaptively refined during solution until a user-specified accuracy
tolerance is met. In time dependent problems, meshes will be refined where
necessary, and un-refined where no longer required, so that mesh density will
follow moving fronts.
Problem Descriptors
FlexPDE uses a sophisticated grammar-based input format, which allows problem
descriptions to be written in a compact and readable form, following very
closely the mathematical description of the equations and parameters. The
problem domain is specified by walking the region boundaries, attaching
boundary conditions as appropriate.
Graphic Output
Graphic output can be requested for any function of independent and dependent
variables and constants. Available graphic formats include contour plots,
surface plots, elevations (line-outs), vector fields, and displaced meshes.
Arbitrary function values, including area and surface integrals, can be
reported on any plot, and a summary page can be written with reports of
arbitrary function values.
Prices:The following articles can be bought now directly from the supplier.
FlexPDE 2D
Parcel delivery / plus forwarding expenses
English
WIN95 / WIN98 / WINNT / WINXP / WIN2000
| EUR 1.076,00 |
FlexPDE 2D+3D
Parcel delivery / plus forwarding expenses
English
WIN95 / WIN98 / WINNT
| EUR 2.674,00 |
