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Additional tasks to unit “Finite Element Method”
Task 1. Read the text below. What can the word “element” in the term “finite element method” mean?
Depending on the author, the word «element» in «finite element method» refers either to the triangles in the domain, the piecewise linear basis function, or both. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear. On the other hand, some authors replace «piecewise linear» by «piecewise quadratic» or even «piecewise polynomial». The author might then say «higher order element» instead of «higher degree polynomial». Finite element method is not restricted to triangles (or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g. ellipse or circle).
Examples of methods that use higher degree piecewise polynomial basis functions are the
hp-FEM and
spectral FEM.
More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve approximate solution within some bounds from the ‘exact’ solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:
- moving nodes (r-adaptivity)
- refining (and unrefining) elements (h-adaptivity)
- changing order of base functions (p-adaptivity)
- combinations of the above (hp-adaptivity).
Task 2. Match the words 1-to their definitions.
| 1. mesh
2. polinomial
3. curvilinear
4. quadrilatteral
5. piecewise
|
E. a regularly spaced array of points in a square array, i.e., points with coordinates (m, n,…)
A. a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients
D. having a graduate smooth bend
B. having four sides like a square
C. composed of some number of segments defined over an equal number of intervals, usually of equal size |
Task 3. Give the definitions of the words: triangle, domain, primitive, higher order element, hexahedron, prism, pyramid, circle, bound, adaptivity.
Task 4. Work with a partner to complete the word families in the table below.
| Verb |
Noun |
Adjective |
|
utilize
refer
modify
depend |
adaptivity
bound
solution |
popular
restricted
defined
approximate
|
Task 5. Compose your own sentences using the words from the table.
Task 6. Ask the questions using the words below.
- popular are what the techniques most?
- refer the can word to “element” what?
- to method the is element is not, triangles restricted finite it?
- adaptivity what utilize mesh does?
- be with high what order defined can shapes?
- functions of piecewise what some basis higher methods use degree examples polynomial are that?
Task 7. Work with the partner to answer the questions in Task 6 and discuss the topic.
Task 8. Write an essay on the topic “Finite Element Method”. Use your knowledge of mathematics and information offered in the unit “FEM”.
Ссылка на первоисточник:
http://rustest.ru/
