Finite element simulation - Reports

The value given by Abaqus is about 3.315 Hz, and it is very close the numerical result.

The next step involved performing a static analysis, which was a straightforward process in Abaqus. For the

dynamic analysis, it was necessary to provide the software with the natural frequencies, which were

determined through a modal analysis. In this context, the load magnitude was irrelevant, as the vibration

modes (eigenvalues) are intrinsic properties of the system and independent of the applied load. For this

48

analysis, only five vibration modes were considered, excluding those outside the frequency range of

180 1 = 16.546 .

(lower limit) and (upper limit). Reporting Abaqus data: Now, from the

1 = 0.07526 .

modal analysis the same value can be obtained at the required frequency:

The modal analysis revealed that 122 Hz is very close to one of the five natural frequencies of the structure.

As a result, only this frequency was considered in the final analysis, while the contributions of the other four

frequencies were neglected. 48 180 .

Once obtained eigenfrequencies it’s possibile to plot the FRF, here cut from to All the peaks

122

are in correspondance of a specific eigenvalue and, as previously mentioned, is very close to one of

them.

Discussion of results th

Looking at the results of the 18 node displacements in the static and dynamic configuration, it can be

observed that at 122 Hz that specific point does not undergo a great displacement. In the static case ,

indeed, there are at least two order of magnitude of difference between the dynamic value.

For the analytical estimation of displacement, the following formula, based on FRF and modal approach,

con be exploited:

4

( )

42

= = 0.0852

4 2 2 2

√(1 − ) + (2 )

4 4 4

Where: 2

= 123.54 = = 776.2 = 0.6845 ∗ 3000 = 2053.6

; ;

4 4 4

4

= × 0.880017 = 0.075

1 4

The value which multiplies comes from Abaqus and represents the coordinates of the node 18, in x-

4

direction, at 123.54 Hz.

ξ

is the ratio between the load frequency and one of the natural frequency of the structure, 123.54 Hz; is

= 0.02.

the ratio between the damping of the structure and the critical damping, assumed Solving the

= 0.075 = 0.07526 ,

equation is obtained. The analytical solution given by Abaqus is: very

1 1

close to the numerical one. There is an error of 0.3%, which can be neglected and probably generated by

approximations, the model con be considered realistic and correct.

The following images illustrate the comparison of reaction forces in the horizontal direction (direction 1)

between the static and dynamic cases. A notable difference is observed in the direction of these forces. In

the static case, the concentrated load is positive along the x-axis, leading to an expected opposite reaction

force at the hinge at the base of the column.

In the dynamic configuration, the sinusoidal nature of the applied load results in upper and lower limits of

= −3000 = 3000 ,

and respectively. Over time, the structure experiences every force magnitude

123.54 ,

within these limits. At a frequency of the load is applied from right to left, resulting in a positive

= 1.

reaction force. This force is further amplified due to the amplification factor

Report N°5: Convergence Analysis

Scope of work

The aim of this study is to investigate how the size and shape of the mesh influence the accuracy of results in finite

element method analysis and to explore strategies for optimizing mesh parameters to achieve precise outcomes. The

accuracy of the analysis is, of course, contingent upon the acceptable tolerance levels. Various approaches to mesh

optimization will be evaluated, with the effectiveness of each method compared to determine the most suitable

approach for specific scenarios.

Description of the model

• Geometry: the element is a square plate with a centred hole, due to its symmetry we can just analyse a

= 20 = 7 , =

quarter of the shape. The side is and the inner radius of the hole is its thickness is

1 so we can use shell element.

7

= 10 = 0.3,

The element is pretty rigid, with a standard Poisson’s ratio the material is unknown.

• Boundary conditions: as symmetry has been imposed, it’s important to use the right boundary conditions

that enable the right relative movement between the symmetrical partitions of the element. In this case two

rollers: the one on the left only allows the sliding along the vertical direction indeed the vertical roller lets the

element move only right word (of left word).

• = 1

Load: there’s just one load applied on the inner surface, on the radius, and it’s a pressure.

• Mesh: the primary focus of this analysis is the impact of mesh size and element configuration on FEM results.

Mesh size can be adjusted through three distinct refinement techniques:

1. H refinement: In H-refinement, the characteristic dimension of each mesh element—denoted as h—

1/2

is modified. This dimension is represented by L for beam or bar elements, for solid 2D or shell

1/3

elements, and for 3D solid elements. Starting from an initial mesh, the number of nodes

increases either uniformly or non-uniformly without altering their original positions. Importantly, the

polynomial order of the elements remains unchanged.

2. P refinement: In P-refinement, the mesh is enhanced by increasing the polynomial order of the

elements. This is achieved by adding mid-nodes to existing elements, thereby enhancing the

element's ability to capture complex gradients.

3. R refinement: R-refinement, or rearrangement, involves repositioning the nodes of the mesh while

keeping the total number of elements and nodes constant. This method allows for a more optimized

distribution of elements based on specific areas of interest in the model.

Results

Following are presented the results of the same analysis performed by simply changing mesh size, order of polynomial

and also the type of integration, with the aim to increase precision and obtain convergency.

The results for the first two cases are presented using the same mesh, one with reduced integration and the other

without. Both cases are deemed unacceptable for different reasons. Reduced integration results in a lower number of

nodes than required, causing the results to be averaged at the center of the nodes. While it is reasonable to expect

the highest stress to occur at 45° from the x-axis, this is not observed, likely because the two elements in question are

larger than the adjacent ones, leading to an averaged lower stress value.

In the second case, the results still do not satisfy the expected conditions. Distinct colors appear between two

adjacent elements along the boundary, which is unrealistic. This issue arises due to the averaging of the results.

Therefore, a refined mesh is necessary, such as the use of quadratic elements, to achieve more accurate results.

Switching from reduced to full integration, the results with averaging show a good linearity within each element. A

simple analysis of stress and displacement magnitudes reveals that in regions with less material, both stresses and

displacements are higher. Specifically, at the angle, the Von Mises stress is negligible due to the reduced material

volume in that area.

Discussion of results

The primary objective of mesh refinement is to obtain more accurate results. However, in the absence of an exact

solution, a stopping criterion is needed, which can be defined by the difference between two consecutive results. This

process is known as convergence analysis. In this case, convergence analysis was performed by progressively reducing

1 0.5 0.25 ,

the mesh size from to and using linear elements.

The previous images show the differences of the same simulation with different mesh sizes, without averaging:

:

1. in the first attempt we have 85 nodes and 64 elements using CPS4 elements;

. :

2. by decreasing the element size we obtain: 310 nodes, 270 elements again with CPS4 elements;

. :

3. the last one mesh is made by 1134 nodes and 1054 elements.

ℎ

In a general case the discretization error, that measures the speed of convergence, is proportional to where p is

the order of polynomial, instead h represents the characteristic dimension of the element size, it can be differently

computed if we have a regular mesh or not, for a broad case we can calculate it as:

1

ℎ=

√

= 2 2, = 3 3

Below is reported the convergence analysis of this case:

Mesh size N h h^p stress error p

1 64 0.125 0.125 3.3862 12.13% 1

0.5 270 0.060858 0.060858 3.6366 5.63% Phi inf

0.25 1054 0.030802 0.030802 3.733 3.13% 3.8537

3.8

3.75

3.7

3.65

3.6

3.55 y = -3,7166x + 3,8537

R² = 0,998

3.5

3.45

3.4

3.35 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

If the analysis is performed using quadratic elements, we can expect an higher speed of convergency due to an higher

order of polynomial. Using same element size the one which changes is just the number of nodes which increases to 4.

Here are reported the results of the analysis:

Mesh size N h h^p stress error p

1 64 0.125 0.015625 3.7663 1.69% 2

0.5 270 0.060858 0.003704 3.8159 0.39% Phi inf

0.25 1054 0.030802 0.0000949 3.8268 0.11% 3.8309

3.84

3.83

3.82

3.81

3.8

3.79 y = -4.1338x + 3.8309

R² = 0.9999

3.78

3.77

3.76 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Report N°6: Plasticity

Scope of work

The objective of this work is to investigate the plastic behavior of materials, specifically focusing on the

permanent deformation that occurs under plasticity. Plastic deformation begins when the material’s stress

state, particularly under multiaxial loading, exceeds the yield limit. To differentiate between plastic and

, = 0

elastic regions, a yield function, has been defined such that indicates the onset of plastic

< 0

deformation and persists in the plastic region until failure. In contrast, corresponds to the elastic

> 0

region. The condition is not feasible, as it would imply that the stress state has exceeded the

material's yield strength. Under uniaxial stress, this relationship is linear and all the data can be obtained by

−

the curve finding the yield stress value with numerical approximations. In a more general multiaxial

state of stress this is no more true and thre