The Method of Generalized Influence Functions – Adjoint Sensitivity Analysis for the Effective and Efficient Numerical Design of Structures with special Application for Reliability and Uncertainty Modelling
A continuation of the topic is currently taking place in the second project phase.
Sensitivity Analysis deals with studying the effects of varied input parameters to the variation of output parameters of any kind of system or model. It is an essential part of system theory. In view of the present SPP the importance of Sensitivity Analysis is twofold: to identify and to rank the significance of parameters in the modelling process and (ii) to explore and to quantify the effect of modifying parameters for the numerical design and optimization of structures, as well as both aspects may be treated in the presence of uncertainty. There exists an enormous amount of work about almost all subjects of sensitivity analysis which may be divided into the two different aspects as introduced above. From the methodological point of view there is a third aspect which deals with the adjoint variant of Sensitivity Analysis which by itself is basically well introduced and understood. In particular, the adjoint Sensitivity Analysis appears to be identical to Green’s function method also known as the method of influence functions (or lines for beams as a specific structure). It seems, however, that there is only very few work in the latter direction and even less to combine it with each or even both of the first two aspects.
The goal of the project is to develop the method of generalized influence functions as an effective and practically relevant mean to determine and to analyze the sensitivity of structures with respect to all kinds of design and model parameters. In particular, the method is intended to serve uncertainty analysis with sensitivity information and detailed analysis of error propagation. The theoretical basis is the identity of the influence function method with the adjoint version of design sensitivity analysis. There are two most innovative aspects: (i) The traditional technology of influence functions and surfaces originally developed for the analysis of the deflection and the stress resultants of beam and plate structures under varied loads will be generalized for modern numerical analysis, any kind of structures, any kind of responses (also others than deflections and stress resultants) and any kind of parameters (also others than load position and intensity), (ii) The method will serve with sensitivity information for spatial continuously distributed parameters for the effort of one additional calculation.
Summary of the Results
The research of the first period was focused on the relationship between adjoint sensitivity analysis and the method of influence functions (general term for influence lines or areas). Based on the knowledge regarding the equivalence between the adjoint variable - an intermediate result of adjoint sensitivity analysis - and the influence function, the traditional method of influence functions can be generalized for different responses, parameters, structures and types of analysis (e.g. geometrically non-linear). The result of this generalization is an engineering tool for the numerical determination of sensitivities.
The main components of adjoint sensitivity analysis were analysed for practical relevant responses and parameters. To investigate the parameter influence the formal equivalence between virtual and adjoint work (performed by the pseudo-load on the influence function) was utilized.
The investigations showed that the parameter influence can be interpreted as a force or stress measure, which interacts with its energetic conjugated displacement or strain measure of the influence function. While the classical approach is limited to the energetic conjugated pair displacement and force, further quantities are necessary for generalized sensitivity analyses. E.g. in the case of a beam these are the curvature of the influence function, which describes the influence of the response, and the pseudo-moment, which in turn reflects the parameter influence. Furthermore, it was discovered that the adjoint variable is not sufficient to generate the complete influence function for some responses. This observation is especially important for stress resultants and support reactions. The characteristic discontinuity (kink or jump) of their influence function can be only generated by combining the adjoint variable with other parts of the sensitivity equation. Hence, the relationship between adjoint sensitivity analysis and influence functions is not limited to the adjoint variable as previously stated.
The strength of the classical method of influence functions is its ability to provide understanding of the system in a visual form. This aspect was also incorporated within the method of generalized influence functions. Means to visualize the final and intermediate results of the analysis were developed. Figure 1 e.g. summarizes the sensitivity analysis of the normal force in the bridge main girder w.r.t. the Young’s modulus of the girders.
Finally, the new approach was applied within different uncertainty analyses. Due to the mechanical interpretability of the sensitivity components e.g. it becomes possible to trace how parameter variations actually affect the mechanical system in detail and how this in turn affects gradient-based global sensitivity measures.
Essential project findings
- Development of the theory of the generalized method of influence functions for first order, local sensitivities
- Determination of adjoint fields, pseudo loads, sensitivity maps, discrete and continuous for many different parameters and response functions
- Sound and quality ensured implementation in KRATOS under open source license
- Development of a Python shell to establish interfaces to other software ready to be used by other groups of the SPP
- Development of visualization tools and first steps towards practical relevant usage of sensitivity data as a new engineering approach to design
- Singular value decomposition of gradient information for effective design update; special application for constrained design update and structural optimization
- Application of generalized influence lines for uncertainty analysis
Prof. Bletzinger
Prof. Dr.-Ing. Kai-Uwe Bletzinger
TU München
Ingenieurfakultät Bau Geo Umwelt
Lehrstuhl für Statik
Fußeder, Martin; Wüchner, Roland; Bletzinger, Kai-Uwe: The Method of Generalized Influence Functions – Adjoint Sensitivity Analysis for the Numerical Design of Structures Under Uncertainty. 14th Virtual Congress WCCM & ECCOMAS 2020, 2021
Fußeder, Martin; Wüchner, Roland; Bletzinger, Kai-Uwe. (2021). Sensitivitätsanalyse mit verallgemeinerten Einflussfunktionen zur Tragwerksbewertung bei Modellparametervariationen/Sensitivity analysis with generalized influence functions for the treatment of model parameter variations in structural analysis. Bauingenieur. 96. 191-200. 10.37544/0005-6650-2021-06-33.