Lecture 32: FE Mesh Genera3on. APL705 Finite Element Method

Size: px

Start display at page:

Download "Lecture 32: FE Mesh Genera3on. APL705 Finite Element Method"

Lorraine Fisher
5 years ago
Views:

1 Lecture 32: FE Mesh Genera3on APL705 Finite Element Method

2 FE Stages The en3re finite element process can be divide into three main groups of ac3vi3es 1. Pre-processing 2. Formula3on and solu3on 3. Post-processing The first part involves the geometry crea3on and discre3za3on of the problem domain The bulk of the second stage is concerned with element formula3on, assembly of s3ffness matrices and the most important numerical solu3on of the system of equa3ons. In the last stage, we present and analyse the results obtained from stage two.

3 Preprocessing Geometry crea3on for the problem domain Crea3on of the nodal and elemental data, coordinates, connec3vity, boundary condi3ons, loading data and material proper3es This ac3vity is known by a common name Mesh Genera3on. As the number of elements and nodes grow in real world problems, it becomes unmanageable to handle such data in any stage of FEM. Therefore in modern FE sotware programs, we find very evolved Pre-processors / mesh generators which facilitate modeling and solving complex problems.

4 Mesh Genera3on The basic purpose of mesh genera3on process is to create element connec3vity data and nodal coordinates by taking some key points as inputs. For convenience of demonstra3on we can consider a two dimensional domain. Consider the 2D problem domain shown below a quarter of a circle. This we call the region. The corresponding block is shown as rectangular area which helps us to generate node numbers Note that the region 4 is a void and corresponds to a line in problem domain

5 Genera3on of Node Numbers The strategy of node numbering is illustrated using an example considered previously. The merger of edges and and arriving at the final node numbering as shown below is key to a consistent node numbering.

6 Genera3on of Coordinates and Connec3vity For the purpose of genera3ng coordinates and connec3vity, we can use the isoparametric master element developed earlier with 8 nodes and 8 shape func3ons. Other ac3vi3es associated with it are Mesh Plo]ng Data handling and edi3ng

7 Post-Processing This is an important task ater obtaining the results by solving the system of equa3ons. In most present day sotware programs both pre and post processors are part of the GUI. Plo]ng of deformed shape / mode shape Gives an idea of how the structure behaves under the applied loads Contour plo]ng different types of results can be graphically visualised using either symbol plots or colour plots. Displaying nodal and element values of variables Calcula3on and display of secondary variables and their analysis

8 Mesh Genera3on and Tests While genera3ng the mesh, a`en3on is necessary for the correctness and accuracy of the results to be achieved. It is necessary to consider the convergence aspect of FE approxima3on right in the beginning Also, rate of convergence becomes important in large and complex problems A check on the robustness of solu3on procedure and algorithms The correctness of programming and implementa3on

9 Convergence Criteria We know that FE method is an approxima3on of the exact con3nuum solu3on. The convergence test decides how closely the two solu3ons agree with each other. The approximate solu3on u should tend to the exact solu3on u when the size of the element h approaches zero, i.e u u' = O(h q ) Ch q Where h is sufficiently small, q>0 and C is posi3ve constant. This must also be true for all the deriva3ves of u The value o q indicates the order of convergence

10 Convergence Criteria The approxima3on should be such that, it sa3sfies both consistency and stability condi3ons u u' = O(h q ) Ch q The consistency requirements: As the size of the element h tends to zero the approximate equa3on (model) will represent the exact differen3al equa3on and boundary condi3ons. The stability Condi3on: It means that the solu3on of the discre3zed system be unique and avoid spurious inclusions that may pollute the solu3on itself. a = K 1 f Which simply mean that the s3ffness matrix be non-singular

11 Patch Test It is a procedure to test the consistency requirements in FEM The basic principle of this test is based on the idea that if we consider sufficiently small domain of size, 2h, we can expand the unknown func3on u and its essen3al deriva3ves appearing in the weak formula3on into a Taylor series Around point i we require that with p 2 The FE approxima3on should reproduce exactly the problem posed for any linear forms of u as h tends to zero u = u i + u x u x = u x The patch test tests only the sa3sfac3on of the differen3al equa3ons but not the boundary condi3ons. i i x + u y +!+O(h p 1 ) u y = u +!+O(h p 1 ) y i i y +!+O(h p )

12 Element Quality Checks As we are assuming the solu3on to be approximate, we need to consider quality of elements in the first place for improving the accuracy of the solu3on obtained. Some examples are shown here graphically Common quality checks test for Skewness, Aspect ra3o, Warpage and Jacobian ra3o

13 Element Quality Parameters Skewness: It is defined as the angle (90-α) between the vectors for triangle and a quad element are shown. Aspect ra3o : This is the ra3o of maximum length of side (dimension) to to minimum side-length of an element Warpage: warpage of 2D elements is found by dividing a quad element into 2 triangular elements and finding the angle between the plane of the two triangles. It is done again with dividing the other corners. The maximum of the two angles is the warpage. Jacobian: The Jacobian ra3on is the measure of the devia3on of the element form the ideal shape element (master element). This ra3o ranges from -1 to 1 where 1 corresponds to an ideal element. Acceptable value is >0.6

FEM Convergence Requirements

19 FEM Convergence Requirements IFEM Ch 19 Slide 1 Convergence Requirements for Finite Element Discretization Convergence: discrete (FEM) solution approaches the analytical (math model) solution in some