Master students

The design of tensegrity structures is still a challenging task despite long research on the topic. Recently, an elegant approach for design of self-stressed tensegrity structures was proposed by Wang et al. (2021). This approach yields the force density distribution and the geometry for a tensegrity structure based only on the topology of cables and trusses in it and additional constraints on the force densities. These force densities provide the necessary stability of the structure. In the core of the approach, a rank minimization problem of the force density matrix is solved. For this purpose, special relaxation of the rank minimization problem are used, e.g. a simple trace minimization.

The goal is to apply and improve the described method and analyse its robustness and convergence. Especially the effect of different constraints on the force densities and their effects on the rank minimization problem and the resulting shapes shall be investigated. In case studies of 2D and 3D tensegrity structures, also with a rank deficit beyond d+1, relations between the rank minimization problem and structural properties of the topologies can be explored.

The specific tasks are

• Literature study on form finding of tensegrity structures and the force density method
• Learning basics of convex optimization and CVX package in Matlab
• Implementation of form finding of tensegrity structures based on rank minimization
• Case studies of 2D and 3D examples to investigate the robustness and convergence
• Study the influence of the constraints between force densities

Recommended fields of interest

Form finding, tensegrity, force density method, Matlab, optimization

Literature

Wang, Y., Xu, X, and Luo, Y.: Form-finding of tensegrity structures via rank minimization of force density matrix. Engineering Structures 227 (2021), doi: 10.1016/j.engstruct.2020.111419

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Beispiel mit einer komplexe Last-Verschiebungskurve.

The calculation of the load-displacement curve of a structure can take a lot of computing time depending on the complexity of the path to be traced. For the path-following of complex equilibrium paths where different critical points occur, arc length methods are particularly suitable. Here, the correct calculation of the load-displacement curve depends on the arc length chosen at the beginning of the calculation. If the initial arc length is too large, the path cannot be followed correctly, especially in sharp curves.

To eliminate the dependency on the size of arc length and to reduce the computation time, an adaptive arc length method based on a subsequent refinement of the arc length shall be developed and implemented. The arc length shall be automatically selected smaller in critical areas like sharp curves. The initially coarse calculation of the equilibrium path also allows a parallel calculation of the subintervals.

The specific tasks are

• Literature research on arc length and multigrid methods.
• Implementation of the adaptive arc length method in Matlab/Ikarus
• Comparison of the performance and efficiency of the implemented method with a standard arc-length method using various examples

Recommended fields of interest

Nonlinear finite elements, parallelization, implementation in Matlab/C++

Literature

[1] H.M. Verhelst, M. Möller, J.H. Den Besten, A. Mantzaflaris, M.L. Kaminski, Stretch-Based Hyperelastic Material Formulations for Isogeometric Kirchhoff–Love Shells with Application to Wrinkling, Computer-Aided Design 139 (2021).

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Gleichgewichtspfad mit kritischen Punkten

Critical points are of utmost engineering significance due to the special physical characteristics of structural behavior in their vicinity. Limit points initiate snap-through of a structure and at bifurcation points the structure begins to buckle. With standard path-following methods, these critical points can be estimated but not calculated exactly, since only individual points of the equilibrium path are determined, which are not necessarily critical points.

There are several methods that can be used to calculate critical points, such as limit and bifurcation points, exactly. Within this thesis, three of these methods should be implemented and compared with respect to efficiency and performance. These are the method of extended sytems [1], the minimally augmented method [2] and bisection methods [3].

The specific tasks are

• Literature research on methods for the exact calculation of critical points
• Implementation of the three methods in Matlab/Ikarus
• Comparison of the methods in terms of efficiency and performance using various examples of limit and bifurcation points

Recommended fields of interest

Nonlinear Finite Elements, stability analysis, programming in Matlab/C++

Literature

[1] P. Wriggers, W. Wagner, C. Miehe, A quadratically convergent procedure for the calculation of stability points in finite element analysis, Comput. Methods Appl. Mech. Engrg. 70 (1988) 329–347.
[2] J.-M. Battini, C. Pacoste, A. Eriksson, Improved minimal augmentation procedure for the direct computation of critical points, Comput. Methods Appl. Mech. Engrg. 192 (2003) 2169–2185.
[3] J. Shi, Computing critical points and secondary paths in nonlinear structural stability analysis by the finite element method, Comput. Struct. 58 (1) (1996) 203–220.

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“Project Chrono” is an open source multi-physics simulation engine. Among many other features it contains a module for solving nonlinear finite element problems as well as contact problems. The aim of this thesis is to explore the capabilities of Project Chrono and compare its features with the state of the art. Therefore, scientific literature needs to be studied and tests need to be run with Project Chrono. Depending on the student’s interest, Project Chrono can also be compared to commercial software codes (e.g. ANSYS or LS-DYNA) or an interface can be implemented to include it in the institute’s finite element code.

The specific tasks are

• Review of publications related to Project Chrono’s finite element module
• Investigation of the abilities of Project Chrono with focus on finite element simulations and contact mechanics by running benchmarks with Project Chrono
• Optional: Compare results with commercial software codes regarding accuracy and efficiency
• Optional: Implement an interface to include Project Chrono in the institute’s finite element code NumPro
• Interpretation, evaluation and documentation of results

Recommended fields of interest

Finite element simulation, contact mechanics

Literature

[1] Project Chrono: https://projectchrono.org/

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The concept of redundancy matrices in discrete truss structures is well-described in literature [1, 2]. The redundancy matrix contains information about the distribution of statical indeterminacy, i.e. of internal constraint. This information is valuable especially in the design and analysis of adaptive structures [3, 4]. Further research towards design and analysis of adaptive surface structures is needed when dealing with structures in continuous space [2], especially in statically indeterminate structural theories like e.g. 2d-elasticity.

The aims of this thesis are (1) investigating the redundancy distribution of discrete structures which are reduced by means of statical and geometrical condensation methods, respectively, and (2) redundancy matrices of structures modeled with different structural models (like e.g. beam-like structures modeled with 2d finite elements).

The specific tasks are

• Investigating the concept of redundancy matrices in truss structures with application in adaptive structures
• Computing redundancy matrices in reduced models by means of statical and geometrical condensation methods, respectively
• Studying properties of redundancy matrices (e.g. eigenspace, idempotence) w.r.t. different structural models

Recommended fields of interest

Structural mechanics, adaptive structures, Maple/Matlab

Literature

[1] Bahndorf, J.: Zur Systematisierung der Seilnetzberechnung und zur Optimierung von Seilnetzen. Doctoral Thesis, Universität Stuttgart, Stuttgart, 1991.
[2] von Scheven, M., Ramm, E. and Bischoff, M.: Quantification of the Redundancy Distribution in Truss and Beam Structures. IJSS, under review, 2020.
[3] Geiger, F., Gade, J., von Scheven, M. and Bischoff, M.: Anwendung der Redundanzmatrix bei der Bewertung adaptiver Strukturen. 14. Baustatik-Baupraxis, Stuttgart, 23-24 March 2020.
[4] Wagner, J.L., Gade, J., Heidingsfeld, M., Geiger, F., von Scheven, M., Böhm, M., Bischoff, M. and Sawodny, O.: On steady-state disturbance compensability for actuator placement in adaptive structures. at – Automatisierungstechnik (2018) 66:591–603.

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The degree of statical indeterminacy n_s is a fundamental property in structural mechanics. While n_s is mainly understood as an integral property, the concept of redundancy matrices provides information about its spatial distribution (see [1] and figure left). This gives valuable insight into the load-carrying behavior being exploitable in design and analysis, as well as in the assessment in terms of safety and reliability [3].

From a safety theoretical point of view, redundant structures are so-called mixed systems (see [2] and figure left/center). This means, that the failure of one element does not necessarily result in the failure of the entire structure. Reliability theoretical quantities, like reliability indices, form a basis for existing semi-probabilistic safety and design concepts, which are applied in practical structural engineering. Such quantities are based on stochastics and reflect defined failure probabilities [2].

The goal of this thesis is to investigate redundant frame structures from a safety and reliability theoretical point of view. For that matter, reliability computations for frame structures shall be conducted and analyzed with regard to existent redundancies. Investigations to what extent redundancy distributions enter semi-probabilistic concepts (e.g. partial safety concept, proofs in ultimate limit states) to date, are desired.

The specific tasks are

• Literature studies on redundancy distribution, as well as safety and reliability theory
• Conduction of reliability computations for selected frame structures
• Analyses of existing safety concepts w.r.t. consideration of redundancy distribution
• Studies on various frame structures and careful documentation

Recommended fields of interest

Structural mechanics, safety and reliability

Literature

[1] von Scheven, M.; Ramm, E.; Bischoff, M.: Quantification of the redundancy distribution in truss and beam structures. Int. J. Sol. Str. 213, pp. 41-49, 2021.
[2] Ditlevsen, O.; Madsen, H.O.: Structural Reliability Methods, Online edition, Wiley, 2005.
[3] Kou, X.; Li, L.; Zhou, Y.; Song, J.: Redundancy Component Matrix and Structural Robustness, Int. J. Civ. Env. Eng. 11, No. 8, pp. 1155-1160.

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