To solve a direct problem means to find analytical or numerical solution for ordinary or partial differential equations that are describing given problem. By having a minimum set of information about the problem, referred to as the condition of uniqueness, one can use powerful tools available nowadays to find a solution for a problem at hand. Inverse problems are defined as those in which some of these data are missing and they should be identified from the known solution of connected direct problem. Sub-group of inverse problems represents parameter identification problems, which are the main focus of this course. The course provides a synergic combination of experimental techniques with numerical simulations and mathematical programming to build a practical procedure based on inverse analysis that should be used for the assessment of unknown material parameters. In the main focus are parameter characterization problems in structural mechanics, although most of the material is applicable with slight modifications also to other engineering fields. Within a structural context discussed in the course, simulations of the experiments are done by finite element modeling (FEM), traditionally by commercial software. The training is oriented to the numerical implementation, and participants will gain practical knowledge in coupling numerical simulations done by commercial codes with optimization routines written by them to have a fully automated procedure for the assessment of material parameters.

After fulfilling this course the students will be able to: -Understand various techniques and iterative algorithms used in the theory of numerical optimization. -Write codes in MATLAB aimed to numerically solve the optimization problems by using first order optimization algorithms. -Write codes for interfacing MATLAB with ABAQUS (commercial FEM software) required for automatic modification of input files necessary for FEM analysis. -Generate fully working inverse analysis procedures by writing all necessary codes and putting them together in order to solve problems of material parameter identification.

Theoretical lectures of the course are giving main concepts of selected, most popular, optimization algorithms that are discussed up to the details of their successful implementation. Detailed theoretical background on optimization theory is omitted in the course, but potentially interested students can be guided through available literature on particular topics. Further on, the concept of inverse analysis is presented and typical types of ill-posedness are covered with appropriate measures for their overcoming. Sensitivity analysis are discussed first in a traditional manner, namely the numerical calculations of first derivatives with respect to sought parameters. Second the propagation of measurements uncertainty is evaluated again through sensitivity analysis by simulating different level and type of measuring noise.

Each particular technique discussed on lectures is practiced through numerical implementation exercises. Further on, simulations of various engineering experiments are performed in a commercial FEM code ABAQUS and the interfaces to be used for practical inverse analysis procedures are written in MATLAB. In final parts of the course a practical procedure based on developed techniques is designed aimed to assess elasto-plastic properties of material starting from indentation tests.

Students should also be familiar with basic programming techniques preferably in MATLAB

Total assigned hours: 65

New material: 30
Elaboration and examples (recapitulation): 20

Auditory exercises: 0
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 4
Review and grading of lab reports: 0
Review and grading of seminar papers: 10
Review and grading of the project: 0
Test: 0
Test: 0
Final exam: 1

Activity during lectures: 0
Test/test: 0
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 50
Project: 0
Final exam: 50
Requirement for taking the exam (required number of points): 30

Inverse analyses with model reduction: Proper Orthogonal Decomposition in Structural Mechanics. Springer Verlag, 2012. Vladimir Buljak; Numerical Optimization. Springer Verlag, 2000. Jorge Nocedal and Stephen Wright.; Inverse problem theory and methods for model parameter estimation. SIAM, Society for Industrial and Applied Mathematics. A Tarantola, 2005.