Buchkapitel

Zusammenfassung

The contribution takes up the concept of preventing locking a priori in the theory of thin-walled structures instead of curing it during discretization. After briefly summarizing the successful concept for avoiding transverse shear locking through reparametrization of primary variables for beams, plates and shells we concentrate on the same approach for preventing also membrane locking. Here, we describe first steps referring to the plane curved Bernoulli beam as a conceptual proof for the new method. Inspired by the so-called Mixed Displacement variational method we discuss three variants of replacing primary displacement parameters by alternative variables.

Zusammenfassung

Concrete double-curved shell constructions have been used in architectural design and building constructions since the beginning of the twentieth century. Although monolithic shells show a high stiffness as their geometry transfers loads through membrane forces, they have been mostly replaced by the more cost-efficient lattice systems. As lattice systems are covered by planar glass or metal panes, they neither reach the structural efficiency of monolithic shells, nor is their architectural elegance reflected in a continuous curvature. The shells of sand dollars’ – highly adapted sea urchins – combine a modular and multi-plated shell with a flexible, curved as well as smooth design of a monolithic construction. The single elements of the sand dollars’ skeleton are connected by calcite protrusions and can be additionally supported by organic fibres. The structural efficiency of the sea urchin’s skeleton and the principles behind them can be used for innovations in engineering sciences and architectural design while, at the same time, they can be used to illustrate the biological adaptations of these ecologically important animals within their environments. The structure of the sand dollar’s shell is investigated using modern as well as established imaging techniques such as x-ray micro-computed tomography (µCT), scanning electron microscopy and various optical imaging techniques. 3D models generated by µCT scans are the basis for Finite Element Analysis of the sand dollar’s shell to identify possible structural principles and to analyse their structural behaviour. The gained insights of the sand dollar’s mechanical properties can then be used for improving the state-of-the-art techniques of engineering sciences and architectural design.

Zusammenfassung

We present ideas and concepts towards defining a common framework unifying abstract metrics in order to quantify key features of technical load-bearing structures and biological systems. Our aim is to transfer biological concepts to technical systems at this abstract level rather than on the basis of their outward appearance or actual functionality. This means that the biological concept generators for load-bearing structures do not have to be load-bearing structures themselves but may instead achieve rather different functionalities. We intend to carry out this transference by generalizing graph-based abstractions of both technical and biological worlds to allow comparisons to be made at an abstract level. We focus in particular on the intrinsically competing aims of optimality versus multi-functionality and robustness. In this review, we present initial attempts towards defining suitable quantitative measures for robustness to serve as a common ground for studying technical systems and biological systems simultaneously. We discuss generic properties of a ubiquitous signalling network motif and potential relationships to a minimal model for a robust truss structure. These case studies suggest that topological complexity can serve as a common source for a design that is insensitive to perturbations and thus robust in the measures of both worlds.

Zusammenfassung

We present ideas and concepts towards defining a common framework unifying abstract metrics in order to quantify key features of technical load-bearing structures and biological systems. Our aim is to transfer biological concepts to technical systems at this abstract level rather than on the basis of their outward appearance or actual functionality. This means that the biological concept generators for load-bearing structures do not have to be load-bearing structures themselves but may instead achieve rather different functionalities. We intend to carry out this transference by generalizing graph-based abstractions of both technical and biological worlds to allow comparisons to be made at an abstract level. We focus in particular on the intrinsically competing aims of optimality versus multi-functionality and robustness. In this review, we present initial attempts towards defining suitable quantitative measures for robustness to serve as a common ground for studying technical systems and biological systems simultaneously. We discuss generic properties of a ubiquitous signalling network motif and potential relationships to a minimal model for a robust truss structure. These case studies suggest that topological complexity can serve as a common source for a design that is insensitive to perturbations and thus robust in the measures of both worlds.

Zusammenfassung

Plant movements can inspire deployable systems for architectural purposes which can be regarded as ideal solutions combining resilient bio-inspired functionality with elegant natural motion. Here, we first give a concise overview of various compliant mechanisms existing in technics and in plants. Then we describe two case studies from our current joint research project among biologists, architects, construction engineers and materials scientists where the aesthetic movements of such role models from the plant kingdom are analysed, abstracted and implemented in bioinspired technical structures for sustainable architecture. Both examples are based on fast snapping movements of traps of carnivorous plants. The Waterwheel plant (Aldrovanda vesiculosa) captures prey underwater and the Venus flytrap (Dionaea muscipula) snaps in the air. We present results on the motion principles gained by quantitative biomechanical and functional-morphological analyses as well as their simulation and abstraction by using e.g. Finite Element Methods. The Aldrovanda mechanism was successfully translated into a similarly aesthetic and functional technical structure, named Flectofold, which exists in a prototype state. The Flectofold can be used as a façade shading element for complex curved surfaces as existing in modern architecture.

Zusammenfassung

For practical application in engineering numerical simulations are required to be reliable and reproducible. Unfortunately crash simulations are highly complex and nonlinear and small changes in the initial state can produce big changes in the results. This is caused partially by physical instabilities and partially by numerical instabilities. Aim of the project is to identify the numerical sensitivities in crash simulations and suggest methods to reduce the scatter of the results.

Zusammenfassung

For practical application in engineering numerical simulations are required to be reliable and reproducible. Unfortunately crash simulations are highly complex and nonlinear and small changes in the initial state can produce big changes in the results. This is caused partially by physical instabilities and partially by numerical instabilities. Aim of the project is to identify the numerical sensitivities in crash simulations and suggest methods to reduce the scatter of the results. Work has been undertaken to identify sources of sensitivities through parameter studies, to improve existing mathematical formulations, e.g. of contact-impact and material models. Furthermore a tool is developed for generation of code from mathematical descriptions of finite elements with the aim to reduce effort required to make modifications of implementations of FEM models.

Zusammenfassung

The contribution describes a two-dimensional discrete element method with polygonal particles in a two-dimensional setting allowing the simulation of granular as well as quasi-brittle material. Different models for soft contact as well as cohesion between the particles are presented. Emphasis is put on the specific features of the polygonal particles. Simulations are compared to results of small scale experiments with regular particles of steel nuts. In addition the capabilities of the method are demonstrated simulating complex concrete specimens with a distinct heterogeneous microstructure.

Zusammenfassung

The contribution gives an overview on a discrete element model with polygonal particles in a two-dimensional setting allowing the simulation of granular as well as quasi-brittle material response. It briefly describes the basic formulation for geometry and applied contact models in normal and tangential direction, supplemented by friction on a background plate. Special emphasis is put on modeling of cohesion; three different models with an increasing complexity are introduced, namely an overlay brittle beam lattice, a beam with damage and an interface model. Homogenization of the discrete particle response is utilized deriving variables like stresses and strains for an interpretation in the context of classical and micropolar continua. Several numerical examples for different loading scenarios are added, among them the simulation of a quasi-brittle material sample with a heterogeneous microstructure. In addition conceptual small scale experiments with regular particles of steel nuts have been performed; results from tests and simulations for samples with and without cohesion are compared.

Zusammenfassung

In an engineering context, the quality of numerical analyses, like finite element calculations, often is related to the achieved accuracy of selected variables, e.g. displacements, stresses, strains, etc. This point directly motivates the development of error estimators based on goal-oriented norms, directly coupled to the values of interest. Therefore, adaptive finite element methods, can be applied to refine the mesh based on this kind of error measurements and give the optimal mesh for the target variable. Compared to the classical global error estimators the goal-oriented error estimators are often denoted as local methods since they most often are related to local variables. As an example of local error estimation a small elastic plate with a crack under concentrated load is considered. The classical global energy norm controlled adaptive finite element method is showing mesh refinement near the crack as well as near the load concentrations. The goal-oriented scheme for the stress near the crack however is refining the mesh only in the area of interest. Error estimators like these are a supplement to the traditional error estimators measuring the finite element error in the energy norm, the natural norm from a mathematical point of view, e.g. Babuska & Rheinboldt, Verfuerth, Zienkiewicz & Zhu, Ainsworth & Oden and Stewart & Hughes. For elliptic partial differential equations, PDEs, like the linear elasticity problem, error estimators in the energy norm are accepted as a general tool, in particular since they show sometimes surprisingly good results also in estimating the error of certain other variables like displacements or certain stress or strain values. But even in this rather simple case goal-oriented error estimators with specifically designed error norms which hold for a user-prescribed tolerance show distinct advantages over the general framework of the energy-norm. In an even more pronounced way the increasing efficiency of goal-oriented error estimators can be observed in the case of initial boundary value problems. In this case of hyperbolic PDEs, the solution and therefore also the discretization error propagate along problem-specific characteristics. As a consequence dealing with global error estimators often leads to very large effectivity indexes for the error of individual quantities. The in general a priori calculated global stability of the problem, measured in the stability constant, cannot capture a corresponding user-specified norm in an effective way. In general, sharp analytical estimates for the strong stability constant can not be determined a priori. Therefore the lack of standard energy norm estimators causes a loss of efficiency, indicated by very large numbers for the effectivity index. One objective of this paper is to compare global energy-norm error estimators with different goal-oriented error estimators and to show the efficiency of weighted error estimation. The outline of this paper is as follows: For the simple linear elasticity problem we give a short overview on standard error estimation techniques in the energy norm. In a straight forward manner we extend the error estimator to a goal-oriented error estimation technique. This general goal-oriented technique is then specified for selected variables with their own individual error norms. These error estimators are derived for individual quantities, e.g. for local values like stress or strain components or selected displacements or values in small sub-regions of the considered domain. To derive such goal oriented estimates, we will introduce dual problems corresponding to the strong form of the underlying primal problem, see e.g. Eriksson, Estep, Hansbo & Johnson, Becker & Rannacher, and Cirak & Ramm. After this we extend the concept to nonlinear problems, e.g. plasticity, viscoplasticity, and finally to linear elastodynamics. For all considered physical problems one-dimensional and two-dimensional numerical calculations are given to demonstrate the efficiency of the underlying error estimation and adaptive refinement technique. We conclude the paper with some remarks on the numerical implementation.