**An Integrated Solution for Structural Dynamics Simulation, Model Verification, Validation and Updating**

FEMtools Model Updating contains modules for:

**Sensitivity Analysis**- Analyses how changes of parameters influences the structural responses. This information can be used for different applications including model updating.**Mod****el Updating**- Iteratively changes updating parameters to make the structure better match the target responses.**Harmonic Force Identification**- Identifies harmonic loads from operational shapes.**Probabilistic Analysis**- Applies uncertainty to parameters to obtain probability distribution on output responses.**Design of Experiments**- Efficient sampling of the design space.

### Sensitivity Analysis

Sensitivity analysis is a technique that allows an analyst to get a feeling on how structural responses of a model are influenced by modifications of parameters like spring stiffness, material stiffness, geometry etc. Sensitivity analysis can be used for the following purposes:

**What-If analysis**- Study the effect of modeling assumptions on the modal parameters or on other response types.**Variational Analysis**- Find the relation between design variables and responses in the entire design space.**Pretest analysis**- Sensitivity analysis can be used in pretest planning applications like studying the effect of transducer mass loading on the modal parameters.**Identify sensitive and insensitive areas of the structure for given response and parameter combinations**- This will help the analyst to decide which parameters and responses to include in the selection for model updating.**Model updating**- The sensitivity matrix is inverted to find a gain matrix. This gain matrix is multiplied with the difference between predicted and reference response values to find the required parameter change to compensate for this error.**Design optimization**- Find the optimal locations to modify the structure in order to shift modal parameter values or other response types.**Acoustic sensitivities**- Structural sensitivities computed with FEMtools can be exported to acoustic analysis packages where they are used for the calculation of acoustic sensitivities.

Sensitivity coefficients quantify the variation of a response value (e.g. resonance frequency or mass) as a result of modifying a parameter value. The coefficients obtained for all combinations of responses and parameters are stored in a sensitivity matrix. Analyzing this matrix yields information on the sensitive and insensitive zones of the structure. Color graphics are available to visualize these different zones and enable a fast optimization of the parameter selection.

Sensitivity analysis and model updating require that the user selects reference responses and parameters.

Sensitivity coefficients are computed internally by FEMtools using a differential or finite difference method. The possibilities depend on the parameter type and on the element formulation. Alternatively, externally computed sensitivity coefficients can be imported. For example, sensitivities computed using SOL 200 in MSC.Nastran can be imported in FEMtools for model updating.

#### Key Features

- Selection of all element material properties, geometrical properties, boundary conditions, lumped masses, and damping factors as parameters.
- Selection of mass, static displacements, strain, resonance frequencies, modal displacements, MAC, FRFs, FRF correlation functions and operational displacement shapes (ODS) as responses.
- Sensitivity for local and global parameters.
- Internal sensitivity analysis :absolute or normalized sensitivities, finite difference and differential sensitivities.
- Pre- and postprocessing of external sensitivity analysis (Nastran SOL 200).
- Sensitivity and gain matrix analysis.

Structural Responses

The following reference response types can be selected for sensitivity analysis:

- Mass, center of gravity and mass moments of inertia.
- Static displacements
- Strain
- Resonance frequencies
- Individual modal displacements
- MAC-values
- Frequency Response Functions (FRF) values (amplitudes at given frequency)
- FRF Correlation Functions values (signature and amplitude correlation)
- Operational displacement shapes (displacement, velocities or accelerations)

Design Variables

The following parameter types can be selected for sensitivity analysis:

**Material properties**- Young's modulus (isotropic or orthotropic), Poisson's ratio, shear modulus and mass density.**Geometrical element properties**- Spring stiffness, plate thickness and beam cross-sectional properties.**Lumped properties**- Lumped stiffness (boundary conditions) and lumped masses.**Damping properties**- Modal damping, Rayleigh damping coefficients, viscous and structural damper values.

Parameter can be selected at either the local or the global level:

**Local parameters**refer to an individual element.**Global parameters**refer to sets of elements instead of an individual element.

Model Updating

FEMtools Model Updating includes utilities and methods to update finite element models to better match reference targets like test data. The updating methods are based on the use of sensitivity coefficients that iteratively update selected physical element properties (like for example material properties, and joint stiffness) so that correlation between simulated responses and target values improves. Response types can be static displacements, mass, modal data, FRFs, operational data or correlation values like MAC. Parameters that can be updated are all mass, stiffness and damping properties used in the definition of the FE model. The resulting FE model can be used for further structural analysis with much more confidence.

Example applications are FE model validation and refinement, material identification from vibration testing, FE model reduction, damage detection, ...

#### How Model Updating Works

Discrepancies between FEA results and reference data like test data may be due to uncertainty in the governing physical relations (for example, modeling non-linear behavior with the linear FEM theory), the use of inappropriate boundary conditions or element material and geometrical property assumptions and modeling using a too coarse mesh. These 'errors' are in practice rather due to lack of information than plain modeling errors. Their effects on the FEA results should be analyzed and improvements must usually be made to reduce the errors associated with the FE model. Model updating has become the popular name for using measured structural data to correct the errors in FE models.

Model updating works by modifying the mass, stiffness, and damping parameters of the FE model until an improved agreement between FEA data and test data is achieved. Unlike direct methods, producing a mathematical model capable of reproducing a given state, the goal of FE model updating is to achieve an improved match between model and test data by making physically meaningful changes to model parameters which correct inaccurate modeling assumptions. Theoretically, an updated FE model can be used to model other loadings, boundary conditions, or configurations (such as damaged configurations) without any additional experimental testing. Such models can be used to predict operational displacements and stresses due to simulated loads.

#### Model Updating in FEMtools

There are many different methods of finite element model updating. FEMtools uses well-proven iterative, parametric, modal and FRF-based updating algorithms using sensitivity coefficients and weighting values (Bayesian estimation). The process begins with the formulation of an initial FE model using initial values for the update parameters. The FEA results that will be used to check correlation with test are computed using the FE model with the current update parameter values. The model updating method uses the discrepancy between FEA results and test, and sensitivities to determine a change in the update parameters that will reduce the discrepancy. The FE model is then reformed using the new values of the update parameters, and the process repeats until some convergence criteria, analyzed by means of correlation functions, is met.

#### Key Features

- Automated, iterative, sensitivity-based updating procedure.
- Built-in parameter estimators (weighted, least squares, multi-objective) or custom.
- Selection of mass, static and dynamic displacements, resonance frequencies, modal displacements, MAC, FRFs, and FRF correlation functions as responses.
- Predefined and customizable target functions.
- Selection of all element material properties, geometrical properties, boundary conditions, lumped masses, damping factors and excitation forces as updating parameters.
- Generic parameters and responses.
- Weighting of updating parameters and targets expressing user-confidence (Bayesian parameter estimation).
- Constraints on updating parameters (max per iteration, abs max, abs min).
- Possibility to combine different parameter types and response types in a single run.
- Support of parameter relations (linear and non-linear equality constraints).
- Option to re-analyze updated models using FEMtools modal solver for fast, approximate iterations.
- Superelement-based model updating.
- Simultaneous updating of multiple models (MMU).
- Using internally or externally computed sensitivities.
- Automated scaling of sensitivity matrix for optimal performance.
- Automated support of internal and external solvers for static or dynamic re-analysis of updated models.
- Tracking of updating parameters and system responses during updating.
- Dedicated tables and graphics to examine results (e.g. parameter changes, tracking).
- Undo functions and database restoration.
- Regrouping of local model updating results.
- Export of updated FE models..

#### Superelement-Based Model Updating

When working with large FE models, a bottom-up modeling, testing and assembly approach should be considered. This is most efficient if superelements are used to model the parts that do not change. If updating parameters are selected in the residual part (= elements that are not included in any superelement), then only the residual part is updated and combined with the superelements with every iteration.

#### Simultaneous Updating of Multiple Models (MMU)

Multi-Model Updating (MMU) is simultaneous updating of different versions of a finite model corresponding with different structural configurations. For each configuration there is a modal test. For example, solar panels for satellites can be tested during different stages of deployment and for each stage there is a FE model. This provides a richer set of test data to serve as reference for updating element properties that are common in all configurations. Such properties can be, for example, the joint stiffness or material properties. Other examples are a launcher tested with different levels of fuel, or differently shaped test specimens made of a composite material that needs to be identified.

Harmonic Force Identification

From measured harmonic operational shapes, and an updated finite element model, a system of equations can be solved to obtain the excitation forces.

#### Key Features

- Force identification from dynamic response measurements.
- Definition of masks for location of forces.
- Identification of harmonic nodal loads or element pressure loads.
- Export of identified forces

### Probabilistic Analysis

All physical properties are subject to scatter and uncertainty. It is important to assess how this variability of properties propagates in a structure and results in also variability on the output responses. This has applications in robust design (for example Design for Six Sigma - DfSS) but is also used for statistical correlation and probabilistic model updating in case multiple tests have been performed.

#### Key Features

- Apply a statistical probability distribution to physical properties and randomly sample thousands of physical properties using only a few commands (Monte Carlo simulation).
- Re-analysis for each sample using FEMtools or external solvers.
- For dynamic responses, a fast approximate modal solver can be used to significantly reduce the time required to run hundreds of simulations.
- Use all parameter and response choices available for Sensitivity Analysis and Model Updating (see above).
- Postprocess simulations to obtain histogram, mean and standard deviation of output responses.
- Importing and Exporting Monte Carlo Simulation Results
- Probabilistic Correlation with Uncertain Test Data.
- Creating Scatter Plots.

#### Statistical Correlation

Statistical correlation is the graphical and numerical analysis of similarities and differences between point clouds and their statistical derivatives (center of gravity, mean, standard deviation, ...).

Test procedures and results extraction methods are also subject to scatter and uncertainty. Test data should therefore be considered as point clouds that can be compared with similar point clouds obtained from stochastic simulation.

Comparing the position, size and shape of point clouds provides additional insight in the quality of the simulation model and it capacity to represent the true physics of the structure being tested.

#### Probabilistic Model Updating

Probabilistic model updating is about modifying design parameters and their random properties to improve statistical correlation between simulation and test point clouds and their statistical derivatives.

#### Design Improvement and Robust Design

When a validated, and thus realistic, simulation model is available, the design can be improved in terms of product performance and robustness. Using a procedure that is similar to probabilistic model updating, design parameters and their random properties are used to modify position, shape and size of simulation point clouds to satisfy design goals and constraints. In most cases these goals are the translation of specifications related to quality, durability and manufacturing tolerance, and thus overall cost.

### Design of Experiments

Design of experiment (DOE) offers a number of techniques to sample the design space of a problem in an efficient way.

In model updating, DOE techniques can be used to find a set of starting values that result in a better correlation with the reference data as the current starting values. DOE is particularly interesting if the correlation between the initial FE-model and the reference data is too poor to perform a sensitivity-based updating.

#### Key Features

- Choice of standard sampling methods (factorial designs, central composite designs, latin hypercube designs, D-optimal designs).
- User-programmable sampling.
- Re-analysis using FEMtools or external solvers.
- Fast, approximate analysis using modal solver option.
- Ranking of sample responses.
- Applying selected samples to the FE-database.

### User Interface

- All definition, editing and analysis accessible via intuitive menus and dialog boxes or using free format commands for batch processing and process automation.
- Complete electronic documentation.
- Dedicated graphics viewers for model inspection and results evaluation.
- Point-and-click interactive selection.
- Direct access to FEA and test data.
- Unlimited customization and extension using FEMtools Script language.

### Prerequisites

- FEMtools Framework with basic FEA Solvers (included)
- FEMtools Dynamics (included)
- FEMtools Pretest and Correlation (included)

### Options

- Upgrade to FEMtools Full Version (includes FEMtools Optimization)
- NASTRAN interface and driver
- ANSYS interface and driver
- ABAQUS interface and driver
- UNIVERSAL FILE interface and driver
- FEMtools Modal Parameter Extractor
- FEMtools Rigid Body Properties Extractor