Uncertainty Propagation in Mechanical Systems (Stochastic finite elements)
Uncertainty Propagation in Mechanical Systems (Stochastic finite elements)
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Uncertainties are inherently present in mechanical systems, for proper design, it is necessary to properly quantify the randomness in the system and to properly include the uncertainty effects in the design and analysis of such systems.
In such analyses the properties of the system (material, loadings etc...) are represented as stochastic processes instead of discrete values.
A program was implemented in MATLAB to carry out such analysis, it consists of several steps:
Karhunen-Loeve expansion to discretize the stochastic processes into the space domain. From the discretization a stiffness matrix that accounts for the uncertainty of the materials can be defined
Finite element analysis of the structure. This is done using two procedures:
Monte Carlo analysis which gives accurate results but is computationally costly
Intrusive polynomial chaos expansion (PCE) which is computationally cheap
The code can be used to study the reliability of a mechanical system with respect to any failure mechanism
Plate with uncertain material properties displacement analysis
Plate with uncertain material properties displacement analysis
A plate in plane stress under static constant loading is considered. The Young's modulus of the plate is considered uncertain and it follows a normal distribution with a mean of equal to 1 and a standard of deviation equal to 0.2. The plate is analyzed using Monte Carlo (took 250 seconds to run) and the PCE method (took 20 seconds to run.
The plate is solved using the finite elements method.
The performance of the PCE is excellent compared to the Monte Carlo simulations and it gave accurate results as seen.
The histograms represent the probability distribution of the displacement of the right upper corner of the plate.
Settlement of a foundation on a elastic soil
Settlement of a foundation on a elastic soil
This example deals with the settlement of a foundation (represented by a pressure load) on an elastic soil with uncertain material properties (Young's modulus follows a normal distribution with a mean of 50 MPa and a standard deviation of 10 MPa).
The probability of failure of the soil is computed using the PCE and Monte Carlo simulations. The failure is represented by the settlement exceeding 10 cm.
The simulations are done using the finite elements method.
The probability distributions of the displacements are found to be similar using both approaches.
Using both methods the probability of failures were also found to be similar: 0.41%.
The PCE approach took 7 seconds to obtain the results whereas the Monte Carlo simulations took around 700 seconds.
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