what is a Iso parametric element:
Iso parametric element: element that can use the same shape functions to describe the interpolation of displacement in the element and to transform from one co-ordinate system to the other. Same shape functions do multiple things in the element
- Simplifying integration
- Allows for analysis of elements with complex shapes
what is a Axisymmetric element:
Elements in the form of a 2D element, used to analyse three dimensional axisymmetric solids subjected to axisymmetric loading.
Difference between lagrangian and serendipity two dimensional quadrilateral finite elements
- Lagrangian Elements: These elements typically include nodes at each corner and possibly along each edge, plus potentially additional nodes at the element center depending on the order (linear, quadratic, cubic, etc.). They are usually rectangular or square in shape.
* Computational cost:
§ More degrees of freedom due to complete polynomial basis.
§ Higher computational cost in processing time and memory.
* Instabilities
§ Prone to numerical instabilities in stiff problems and fine meshes.
§ Susceptible to ill-conditioned system matrices.
§ Higher risk of volumetric locking in simulations of incompressible materials.- Serendipity Elements: These elements usually have nodes at the corners and on the edges, but they do not typically include internal nodes even in higher-order formulations. They can be formed into more complex shapes, such as rectangles that are not necessarily square.
- Computational cost:
§ Fewer degrees of freedom by omitting internal higher-order terms.
§ Better computational efficiency, especially in large systems.
§ More commonly used, just use more of them
§ P method-> increasing order of elements (not commonly used)
§ H method -> increasing number of elements - Instabilities
§ Generally exhibit fewer stability issues.
§ Reduced complexity helps avoid numerical errors like locking.
- Computational cost:
- Serendipity Elements: These elements usually have nodes at the corners and on the edges, but they do not typically include internal nodes even in higher-order formulations. They can be formed into more complex shapes, such as rectangles that are not necessarily square.
What are the advantages of using a natural coordinate system in the formulation of finite elements?
Vary between -1 and 1
Integrations and calculations all between these values
The same calculation can be applied to any element regardless of the co-ordinate or dimension
Doesn’t matter shape or location (dimensionless)
Shape function for k node in a four node quadrilateral
Sk = 1 at node k
0 at all other nodes
All shape functions add to one
Linear transition?
Shape function for corner node and mid-side node of an eight-node quadrilateral element
Value of one at own node
Value of zero at all other nodes
Quadrilateral curve joining
Types of nonlinearity
- Geometrical
○ Non linearity due to large displacements, large rotations and small strains
§ Example: The bending of a long, slender fishing rod. The rod can bend significantly and rotate without the material itself undergoing significant deformation (small strain).○ Non linearity due to large displacements, large rotations and large strains § Example: Rubber Seal Compression: As a rubber seal compresses, it experiences significant deformation, including large displacements and strains.- Material
○ Material non linearity
§ Example: The plastic deformation of a steel beam under a heavy load. Once the yield strength is exceeded, the steel beam will not return to its original shape. - Contact
○ Non linearity due to contact
Example: The impact between a ball and a rigid wall. The area of contact, pressures, and resulting forces change dynamically during the impact.
- Material
Types of analysis
- Plane strain: long structures with uniform cross sectional details. Assumes that strains in the direction perpendicular to the plane are zero (2D)
- Plane stress: Thin structures where the stress in the thickness direction is negligible. Assumes stress out of plane are zero (2D)
- Axisymmetric: Rotationally symmetric around axis, where loading is also symmetric (2D)
- Solid: Thick components where stresses and strains must be analysed in 3D
- Shell: Thin curved surfaces where the thickness is small compared to other dimensions (3D behaviour with 2D representation)
Finite element and material model for analysis:
- A plastic calculator housing under load from being sat upon
○ Shell, with nonlinear elastic material
Finite element and material model for analysis:
- Floor of a house loaded with furniture. Floor has wooden joists and plywood flooring
○ Beam and shell model, with orthotropic material model
Finite element and material model for analysis:
- Coffee cup loaded with coffee, where we are interested in the stresses where the handle joins the cup
Solid 3D model (as asking for specific stress), linear elastic material
Difference between sub-structuring and sub-modelling
- Sub structuring: Taking something large and modelling specific sections of the model, being built independently, but sharing nodes. This is done so that the models can essentially talk to each other
- Sub modelling: Taking a large model, and overlaying a sub model on top of this to get more information about an area of interest
Boundary conditions
- Symmetry
- Force
- Distributed force
- Pressure
- Angular velocity
- Velocity
Moment (rotation)
Diagram to show boundary conditions and loads for following situations:
- An office chair with rollers, loaded with the weight of one person
Pressure load on seat (for person), frictionless boundary conditions to model rollers, might want to hold one wheel (still get rolling but simulation would still work)
Diagram to show boundary conditions and loads for following situations:
- One shelf in a bookshelf loaded with books
Plane strain elements (beam elements would work also), pressure load (distributed load for beam), shelf held at either side
Diagram to show boundary conditions and loads for following situations:
- Turbine blade in an aircraft engine
○ Centre of rotation defined, angular velocity, 3D solid elements
Bonded contact:
Assumed that the nodes are essentially welded together and cannot move. Allows transfer of force between structures (example: screw) [REQUIRES: Only linear analysis]
Frictionless contact:
Have movement between two surfaces but no friction (example: project) [REQUIRES: Mostly linear analysis, depending on application]
Frictional contact:
Have movement between two surfaces but friction must be overcome (example: project) [REQUIRES: Non linear, decay or non linear behaviour occurring]
Penalty approach
- One node connected to another node, when they go through each other, a stiffness is generated that will pull the nodes back apart again
Newton Raphson method
- Iterative based equation
- Set initial guess
- Find better approximation from initial guess
- Iterate until desired accuracy is reached
Ansys newton Raphson:
- Non linear problem
- Do not yet know the response
- Initial estimate given (closer the estimate, the faster the convergence)
○ Displacement obtained u_1, and sum of all internal forces generated in the system of elements called F_1.
○ If in equilibrium F_1 = F_A - Recalculate next iteration
○ stiffness and displacement u_2, and subsequent F_2 obtained
○ Smaller difference between F_2 and F_A - Iterate until difference between F_i and F_A is zero, or hits tolerance
○ Must converge to tolerance
○ Once achieved, called the force converged solution
○ Tolerance is called the convergence criterion
- Start with zero displacement or displacement from the previous timestep
- Linearize and evaluate the stiffness matrix based on current displacements and other factors like nonlinear material and contact status
- Compute the internal force from the element stresses
- Calculate the displacement increment and add to current displacement vector
- Calculate residual force
- If converged, this force should be lower than the convergence criteria
- If not, repeat
Shape functions
For 3 node triangle
Si = ξ
Sj = η
Sk = 1- ξ - η
Stiffness matrix for three node triangular element
[k^e] = t*A[B]^T [D][B]
