Although a number of computational approaches exist for
solving engineering problems, the meshless modeling method is gaining traction
among scientists and engineers. Other techniques such as finite
difference/volume, finite element, and boundary element have been typically
used to tackle complex engineering problems that often require extensive
meshing; however, meshless methods can be just as accurate and are faster,
simpler, and require far less data storage.
"It hasn't been too long ago that the idea of
numerically solving complex partial differential equations (PDEs) without the
need of a mesh was essentially deemed impossible, except where analytical
methods could be used on simplified problems," says Darrell W.
Pepper,
professor of mechanical engineering and director of the Nevada Center for
Advanced Computational Methods at the University of Nevada-Las Vegas.
"However, over the past decade, meshless methods—being able to solve PDEs
using node points within a problem domain without the need for nodal
connectivity—began to show up in the area of solid mechanics. These quickly
grew into applications involving heat transfer and more recently into fluid
dynamics and related transport areas."
Ease of Use
Meshless methods are uniquely simple, yet provide solution
accuracies for certain classes of equations that rival those of finite elements
and boundary elements, without requiring the need for mesh connectivity.
Meshless also requires no domain or surface discretization or numerical
integration.
A Petrov-Galerkin finite element model that employs local
mesh adaptation is being developed to determine potential wind energy sites
within the state of Nevada. Image: Ncacm.unlv.edu
"The popularity of the meshless approach results from
the ease with which a problem geometry can be established, without the burden
of constructing complex grids, and the simplistic solution technique utilizing
basic geometric concepts, such as radial basis functions," says Pepper.
Pepper recounts a project where he and several colleagues
spent nearly six months creating an acceptable mesh to solve thermal-hydraulic
flow within a nuclear reactor that contained 600 assemblies—over 150 million
nodes were required to complete the project. Such large problems generally
result in extremely demanding computational resources, typically requiring
supercomputers.
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"Alternatively, a meshless method is not restricted to
dimensional limitations," Pepper continues. "An infinite domain can
be modeled (depending on the number of nodes) and run on a PC. The accuracy
achieved, with even a limited number of nodes in a meshless method, compares
closely to solutions obtained using massively refined grids. Having the
versatility of the ubiquitous finite element method and its use of unstructured
meshes (elements), the meshless method is becoming a quick, accurate, and
viable alternative to these more popular, conventional numerical
approaches."
Advantages Abound
Meshless methods include kernel methods, moving least square
method, partition of unity methods, and radial basis functions. Meshless
methods utilizing RBFs create mesh-free algorithms that are significantly
simpler to employ than more standard approaches. Other advantages of meshless
methods include:
• Significantly
reduced costs compared to current, expensive commercial codes for doing complex
analysis
• Computer platform
independence—apps will eventually be written that will enable the method to run
on a table or even smart phone
• Problem class
flexibility—almost any problem that can be described with a set of PDEs can be
solved using the method
• Familiarity of the
method and use within undergraduate curriculum and advanced extrapolation to
graduate level work—simple models have been written using MATLAB, MAPLE,
MATHEMATICA, and FORTRAN, including C++ and JAVA
Efforts are now underway to develop localized meshless
techniques that are exceptionally fast, easy to employ, and can be used in
multiphysics applications.
"Recent work has included modeling indoor air
pollution, including terrorist threats, subsequently resulting in a real-time
response system that can automatically activate preventive measures," says
Pepper. "Another application is in bioengineering, especially in modeling
aortic blood flow, distribution of air and particulates within the lungs,
effects of heat transfer, and structural prosthetics. The method is also now
being used to model wind fields for wind turbine siting—compared to the previous
method of utilizing hp-adaptive finite element techniques to resolve irregular
terrain based on sparse meteorological data."
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