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Very fundamental questions on XFEM
1)XFEM is suitable in simulatng crack propogation problem because using conventional FEM this
would have required repeated meshing.This is what I've read in most introductory chapters regarding XFEM but what is
its application beyond crack propogation studies?
2)Parttion of unity framework:The main idea is to construct basis functions as products of piecewise shape functions
and local enriched basis functions.But my question is how the concept works in exactly smulating the requirement of
the problem?What is sp special of taking the PRODUCT of piecewise shape functions and local enriched functions?
3)To achieve a quadratic or cubic field in conventional finite element - would obviusly result in an increase i
degrees off freedom-but constructing a quadratic or cubic field through XFEM results in a fewer degrees of freedom
compared to conventional XFEM.Is this correct-any justification for the same?
re: fundamental questions on the X-FEM
1) It has been applied to a wide range of "evolving-boundary-value" problems. What I mean by this is problems in which the evolution of the domain is unknown a-priori, and is a part of what is sought in the solution. Examples include sharp phase transitions, fluid-structure interaction, and laser surgery, just to name a few.
2) The product insures that the enriched basis functions exhibit the proper continuity. It is challenging to embed analytical functions into finite element approximations otherwise.
3)This is correct. One can enrich with polynomial fields and require fewer degrees of freedom on the element level than classical finite elements. However, the functions themselves will be higher-order, which in turn requires additional quadrature for integration. So it won't necessarily be more efficient.
Evolution of the domain
Thanks Sir-
As you said,in (1) it is used wherein the evolution of the domain is unknown apriori--that means the progression of the domain is unknown apriori---by apriori -you mean before you begin the problem solving procedure?
Would it be possible to put a simple understandable example/case?
re:evolution of the domain
That is what I mean, yes.
让我们考虑一个简单的例子的一个域n initial flaw. Typically engineers are interested in how this flaw will grow as a function of the loads applied to the domain. Here one may view the growth of the flaw (or crack if you like) as not being known. All we know are the initial conditions.
As the crack grows a new surface is created. Thus, the boundary of the domain is changing.
Further questions on your slides Sir
Thank-you sir--Your slides are very helpful-below we post some more very fundmantal questions- also we metion the page numbers of the slides where the questions are taken from:
Page 4 question:
1)它是提到XFEM可以的地方使用topology is complex-does it mean where the shape of the domain is highly irregular?Then,it is not an "evolving" problem?IS this a common applicaation of XFEM?
Page 9 question:
2)You mention that -"What's common with FE shape functions is that their supports co-incide with a set of finite elements".By supports do you mean the element ends?
3)Page 10-what do the figures exactly trying to say?Does it show the approximation of the field variable?
4)Page 12-sir-here you are speaking of the principle of mesh free methods wherein, you say:"Most mesh free methods begin by considering a spline function with each node"--what do you mean by a spline function?
Is it that the mesh free method is "free" of elements-however nodes do exist?
5)What do the referencs in red font (BKOFK96),etc (on many other pages) denote?
Page 11
6)The summation on the last line of page 11-what does it denote?Does it denote that the sum of the basis functions over all the nodes is 1?
7)What does the second sumation on the last line of page 11 denote?
8)Similarly, what do the three sumamtions on page 12 denote?
Page 12
9)On page 12,you've mentioned that the problem with this approach was that there is no simple way to turn off the enrichment.Let us say we are dealing with a carck propogation problem-does it mean that enrichment need to be turned off when the crack closes-As long as crack exists the enrichment shall exist-right?
10)Suppose, we have a practical problem (say in a material like concrete), wherein the crack starts due tensile cracking of concrete and later due to reversal of stresses the crack tends to close-does it mean that it is in such case where enrichment need to be turned away but this not possible in msh free methods?If so, how is it possible in XFEM?
re: further questions
I don't actually have the slides in front of me so I'll just do the best I can to answer these.
1) X-FEM can certainly be used to model domains with irregular shapes. This is becoming more common.
2) A node is connected to a set of elements. The support of the node concerns the portion of the domain where the nodal shape function is non-zero. This usually consists of the region formed by the union of all the elements connected to the node.
4) A cubic spline would be an example. Here'sa link.
5) On the very last slide for each lecture is a page with references. This is my short-hand for the references. Each capital letter denotes the last name of an author, and the last two numbers denote the year it was published.
9) No, I'm not talking about crack closure here. I'm simply referring to the idea that one would like the enrichment to only be active over a small portion of the domain.
10) Enrichment does not need to be removed for the problem you describe.
Point number 2 and point number 10
Point number 2:
Sir, you said:
"A node is connected to a set of elements. The support of the node concerns the portion of the domain where the nodal shape function is non-zero. This usually consists of the region formed by the union of all the elements connected to the node."
In conventional finite element, a shape function for a node has a value "unity" at the node and zero at all other nodes within the element.
Suppose we have an element with nodes 1,2,3,4
So, the shape function for a node 1 will be non-zero at the node 1 itself and the whole region within the element except nodes 2,3,4 - in this case what is the support for the node 1?
Point no.10:
Sir, you said:
“浓缩does not need to be removed for the problem you describe."
In the example,a crack formed (hence requires enrichment), later crack closed-so crack foes not exist in the domain anymore-so won't we have to remove the enrichment at this instant?
re: points 2 and 10
For the case you mention, the support of node 1 is the entire element. This will also be the support for nodes 2,3, and 4.
If the crack opens and forms two new surfaces, and then compression causes the two faces of the crack to come into contact (closure), then what you should do is not remove enrichment but enforce contact constraints on the surfaces.