Conference Agenda

We present an efficient, robust, objective, singularity-free, and path independent formulation for director fields based on optimization on manifolds. This approach allows for accurate and efficient computations of director fields that arise in geometrically non-linear structural models such as the Reissner-Mindlin shell model, in material models of Cosserat-type and in micromagnetic simulations. In this contribution, we investigate the influence of interpolation on manifolds on locking as well as the application of element technologies, such as enhanced assumed strains and the discontinuous Galerkin method.

The numerical methods are implemented into the open source code Ikarus (https://ikarus-project.github.io/), which enables rapid algorithm prototyping, even for optimization on manifolds, thus highlighting the user-friendly interface of this software.

The pertinent constraint for director fields requires to retain unit length of the director during deformation, which can be satisfied by interpreting the constraint as a restriction on the design space. By transforming the problem from “constrained optimization on an unconstrained space” to “unconstrained optimization on a constrained space”, the structure of the problem is retained, and the design space is reduced. The transformation to an unconstrained optimization problem on a manifold requires generalization of concepts, such as the incremental update of design variables, to account for living on a manifold instead of living in a linear vector space.

For the interpolation on nonlinear manifolds, we utilize the ideas on geometric finite elements presented by Sander (2012) and Grohs (2011). The combination of element technologies such as enhanced assumed strains with the optimization on manifolds approach promises an efficient and accurate solution method for director fields. Numerical examples are presented in the context of micromagnetics, Reissner-Mindlin shells and three-dimensional beams to demonstrate the efficiency and accuracy of the approach.

Sander, O., Geodesic finite elements on simplicial grids. Int. J. Num. Meth. Engng. (2012) 92:999–1025. https://doi.org/10.1002/nme.4366

Grohs, P., Finite elements of arbitrary order and quasiinterpolation for data in Riemannian manifolds. Seminar for Applied Mathematics, ETH Zürich, (2011). https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-56.pdf

Müller, A., Bischoff, M. A Consistent Finite Element Formulation of the Geometrically Non-linear Reissner-Mindlin Shell Model. Arch Computat Methods Eng (2022). https://doi.org/10.1007/s11831-021-09702-7

Recently, the concept of hierarchic structural element formulations has been developed in the group of the authors with a focus on shear deformable Reissner-Mindlin shell formulations [1], [2]. Via reparametrization of the kinematic variables, these formulations possess distinct degrees of freedom for transverse shear. One effect of this hierarchic parametrization is that the resulting elements are intrinsically free from transverse shear locking.

However, the hierarchic structure can also be exploited for an intrinsically selective mass scaling, i.e., a scaling down of the high shear frequencies, which limit the critical time step although being of minor importance for the structural response, while keeping the low bending dominated branch of the frequency spectrum unaffected. This stands in contrast to conventional mass scaling for shear deformable elements, where total rotational inertia is scaled and, therefore, also bending frequencies are manipulated.

In linear kinematics, the hierarchic parametrization leads to an additive structure throughout the kinematic equations, i.e., a clear separation between a Kirchhoff-Love type bending part and an additional shear part. For nonlinear shell kinematics, the assumption of only small shear rotations was made to preserve this additive structure [3].

In this contribution, we present recent investigations on intrinsically selective mass scaling with hierarchic isogeometric structural element formulations and discuss the effects of transverse shear parametrization in transient problems. Additionally, we critically discuss the necessity of a fully nonlinear treatment of shear deformation parts as described in [4].

References:

[1] R. Echter, B. Oesterle and M. Bischoff, A hierarchic family of isogeometric shell finite elements. Comput. Methods Appl. Mech. Engrg., Vol. 254. pp. 170-180, 2013.

[2] B. Oesterle, E. Ramm and M. Bischoff, A shear deformable, rotation-free isogeometric shell formulation. Comput. Methods Appl. Mech. Engrg., Vol. 307, pp. 235-255, 2016.

[3] B. Oesterle, R. Sachse and E. Ramm and M. Bischoff, Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization. Comput. Methods Appl. Mech. Engrg., Vol. 321. pp. 383-405, 2017.

[4] Q. Long, P. B. Bornemann and F. Cirak, Shear-flexible subdivision shells. Int. J. Numer. Meth. Engng., Vol. 90, pp. 1549-1577, 2012.

The critical time step in explicit transient analyses depends on the highest frequency of the discretized system. In case of thin-walled structures discretized by solid or solid-shell elements, the critical time step, which is a key factor for computational efficiency, is limited by the highest frequencies related to thickness stretch of the elements [1].

Selective mass scaling (SMS) concepts aim at scaling down the highest frequencies, while keeping the low frequencies as unaffected as possible. Most established SMS concepts are designed for discretizations composed of solid or solid-shell elements, as can be seen for instance in [1,2]. They are designed such that at least translational inertia is preserved. Accuracy of these SMS concepts can be increased by extending the construction of scaled mass matrices in such a way that, additionally, rotational inertia is preserved. But this increases computational costs in case non-linear analyses including large rotations, since scaled mass matrices are anisotropic and need to be reassembled during simulation. These additional costs do not pay off in most applications.

In this contribution, we present recent investigations on SMS techniques, which are based on a concept from finite element technology, that is the Discrete Strain Gap (DSG) method [3]. We show that these novel SMS concepts naturally preserve both translational and rotational inertia and possess high accuracy. In addition, having non-linear problem classes including large rotations in mind, we show how to develop efficient isotropic DSGSMS concepts which avoid the need for reassembly of scaled mass matrices.

REFERENCES

[1] G. Cocchetti, M. Pagani und U. Perego. Selective mass scaling and critical time-step estimate for explicit dynamics analyses with solid-shell elements. Computers & Structures, Vol. 127, pp. 39–52, 2013.

[2] L. Olovsson, K. Simonsson und M. Unosson. Selective mass scaling for explicit finite element analyses. Int. J. Numer. Meth. Engng., Vol. 63(10), pp. 1436–1445. 2005.

[3] K.-U. Bletzinger, M. Bischoff und E. Ramm, A unified approach for shear-locking-free triangular and rectangular shell finite elements. Computers & Structures, Vol. 75(3), pp. 321-334. 2000.

For the description of the load-bearing behavior of structures, the degree of statical indeterminacy is a fundamental property that formally describes the number of missing equilibrium equations necessary to calculate the internal forces. The formal definition of the degree of statical indeterminacy as one single number neglects the distribution of statical indeterminacy within the structure. This neglect can lead to situations where a structure, which is kinematic in one direction and statically indeterminate in another direction is denoted as statically determinate. Thus, mechanisms, which are relevant in the field of deployable structures, and possibilities for prestressing are overlooked. The redundancy matrix, first described by Bahndorf (1991), quantifies the distribution of statical indeterminacy in the structure.

The redundancy matrix is an idempotent matrix, meaning that its eigenvalues are either zero or one. Associated with the eigenvalue of one, which occurs exactly in the quantity that matches the degree of statical indeterminacy, the respective eigenvectors span a space of incompatible elongations (von Scheven et al. (2021)). This space matches the description of self-stress states, described by Pellegrino and Calladine (1986). The eigenvectors associated with the zero eigenvalues span a space that includes states where prescribed elongations match the total elongations. In this case, displacements are present without imposing normal forces, even in statically indeterminate structures. The information about states of stress-free displacements and displacement-free stresses can e.g. be used in the decision-making process of actuator placement in adaptive civil structures (Wagner et al. (2018)) and it might also be used in the adaption process of deployable structures (Veuve et al. (2017)) or for maintenance issues like monitoring stresses in certain parts of a structure.

This contribution presents examples of using the redundancy matrix in the design and assessment of civil engineering structures, especially of deployable structures.

References:

Bahndorf, J. (1991). Zur Systematisierung der Seilnetzberechnung und zur Optimierung von Seilnetzen. Ph. D. thesis, University of Stuttgart, Stuttgart.

Pellegrino, S. and C. Calladine (1986). Matrix analysis of statically and kinematically indeterminate frameworks. Int. Journal of Solids and Structures 22(4), 409–428.

Veuve, N., A. C. Sychterz, and I. F. Smith (2017, December). Adaptive control of a deployable tensegrity structure. Engineering Structures 152, 14–23.

von Scheven, M., E. Ramm, and M. Bischoff (2021). Quantification of the redundancy distribution in truss and beam structures. Int. Journal of Solids and Structures 213, 41–49.

Wagner, J. L., J. Gade, M. Heidingsfeld, F. Geiger, M. von Scheven, M. Böhm, M. Bischoff, and O. Sawodny (2018). On steady-state disturbance compensability for actuator placement in adaptive structures. at - Automatisierungstechnik 66(8), 591–603.