1. Dr Xinhui Ma
Computer-Aided Design & Computer Graphics
Division of Engineering and Applied Mathematics
My Research Adaptive Mixed Finite Element Method
T-spline Local Refinement
About Me Publications / News Contact Me
My research interests are in Computer-Aided Design &
Computer Graphics and Finite Element Methods. My recent
research focuses on local refinement of T-splines, NURBS
(Non-Uniform Rational B-Splines) compatible Subdivision
Surfaces and Adaptive Finite Element Methods.
NURBS and subdivision surfaces are two principal
representations of 3D models in engineering (cars, aircraft,
ships) and entertainment (animation and special effects). We
aim to bring the flexibility of subdivision and the accuracy of
NURBS together to engineering and entertainment applications.
T-splines are recently developed generalization of NURBS
technology. T-spline correct the deficiencies of NURBS in that
they permit local refinement and coarsening, and a solution to
the gap/overlap problem. We want to develop efficient local
refinement approaches to speed design and simulation.
Adaptive Finite Element Methods are used to design new
products using optimal simulation by numerically solving partial
differential equations. Adaptive FEM can achieve exceptionally
faster than standard FEM.
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Ainsworth, M. and Ma, X. 2012. Non-uniform order mixed FEM
approximation: implementation, post-processing, computable error
bound and adaptivity. Journal of Computational Physics. 231(2):
pp.436-453.
Ma, X. and Cripps, R.J. 2011. Shape preserving data reduction for 3D
surface points. Computer-Aided Design. 43(8): pp.902-909.
Ma, X. and Cripps R.J. 2007. Estimation of end curvatures from planar
point data. Proceedings of 12th IMA International Conference on
Mathematics of Surfaces XII, Sheffield. Springer-Verlag, Berlin,
Heidelberg. pp.307-319.
Dr Xinhui Ma
School of Engineering,
Computing and Applied
Mathematics
University of Abertay Dundee
Dundee
DD1 1HG
T: 01382 308224
E: x.ma@abertay.ac.uk
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Dr Xinhui Ma is a lecturer of
software application within the
School. He has attended two
EPSRC projects of “Unifying
NURBS and subdivision” and
“Adaptive High Order Finite
Element Methods for
Optoelectronic Devices”, and
published more than ten papers
in international journals and
conferences.
NURBS Compatible Subdivision Surfaces
1. Background
NURBS is a principal representation of 3D model in
engineering and entertainment and extensively used in current
applications. As shown in Fig.2, designers can model complex
3D shapes by manipulating the control points.
Subdivision Surface is a principal representation of 3D model
in films and computer games. Subdivision surface can easily
model arbitrary topological shapes using a single surface.
Fig.2 A NURBS patch and its control polygon
2. Problem and Method
NURBS needs time-consuming ‘stitching’ of separate surface
patches. Subdivision surfaces can easily represent arbitrary
topology without ‘stitching’, but they are not fully compatible
with NURBS at the moment. NURBS Compatible Subdivision
Surfaces can use the flexibility of subdivision approach without
forgoing the significant advantages of NURBS. Existing
methods partially achieve the compatibility in odd degrees. We
aim to fully realize the compatibility of NURBS and subdivision
surfaces in both odd and even degrees, using a refine and
smooth factorisation to insert unequal and multiple knots.
Fig.3 subdivision surface fitting by feature extraction (a) Original meshes
(b) Extracted features (c) Connectivity (d) Fitted subdivision surfaces
3. Applications
NURBS compatible subdivision will bring improvements in 3D
design work flow, so that cars, aircraft, and animated characters
can be modelled more quickly and more flexibly than before.
We applied subdivision surface fitting to dense triangular
meshes as shown in Fig.3. The last model of each row is
presented by a single subdivision surface .
2. Solution
A post-processing scheme is used to approximate high
order non-uniform order mixed FEM. As shown in Fig. 4, the
scheme includes solution of an elementwise local Neumann
problem, local inter-element smoothing and the solution of
an elementwise local Dirichlet problem. Elements with large
errors are processed by h or p refinement for more accurate
approximation .
1. Introduction
hp-Finite Element Method (FEM) is an adaptive numerical
technique for finding approximate solutions of partial
differential equation models arising in mechanics, thermal,
electromagnetic, fluid and environments. hp-FEM can
achieve exponentially faster efficiency than standard FEM.
Mixed FEM gives accurate approximation where the flux or
stress is the primary quantity of interest while the potential
or displacement plays a secondary role. However, there is
no existing non-uniform high order hp version for mixed
FEM. We developed such an adaptive method to achieve
fast and accurate approximation for mixed FEM .
3. Numerical examples
We illustrate the quality and performance of the adaptive hp
mixed FEM for a challenging example of a ten-pole electric
motor, as shown in Fig. 5. The error estimator provides very
satisfactory guaranteed upper bounds on the true error in
both the h and hp adaptive refinement algorithm. The hp
algorithm converges fastest compared with the four h
adaptive algorithms.
Fig. 4 Illustration of post-processing procedure. (a) Original potential
(b) Neumann post-processing (c) Smoothed potential (d) Dirichlet
postprocessing
Fig. 5 hp adaptive mixed FE for a motor (a) A ten-pole electric motor
(b) Geometry (c) hp adaptively refined mesh (d) Convergence
Fig.1 Examples of shape deign by T-splines
1. T-splines
T-Splines are a powerful computer-aided design (CAD) surface
with special properties, which improves upon traditional CAD
technology while retaining compatibility. T-splines allow T
junctions in the control polygons, so that have the following
properties:
• Add detail only where necessary
• Create non-rectangular topology
• Easily edit complex freeform models;
• Maintain NURBS compatibility
2. Local Refinement
This project aims to achieve non-uniform general degree local
refinement schemes for T-splines, so that detailed local
features can be modelled efficiently.
3. Applications
T-Splines are applied in wide areas for design: Jewelry,
Animals, Character, Automotive, Aerospace, Marine,
Architecture, etc. Fig. 1 illustrates some examples designed by
T-splines.