Constraint-Driven Design with Combinatorial Equilibrium Modelling

This paper presents an extension to the Combinatorial Equilibrium Modelling (CEM) design framework. So far, CEM gives the possibility to generate and explore multiple spatial equilibrium solutions in the early conceptual design phase. In addition to the form and force diagrams of graphic statics, CEM introduces a topological diagram, which allows to adjust the connectivity of a spatial strut-and-tie network in equilibrium, its combinatorial inner force state (tension-compression) as well as its metric values (such as element’s length or force magnitudes) in real time. In order to solve more specific design problems within the CEM framework, hence to guide design explorations, constraints-driven adaptation procedures have been developed. This paper describes how these interactive and automatic adaptation methods can be embedded in an overall structural design process containing alternating steps of lateral exploration and vertical adaptation. Moreover, the paper describes how design constraints can be transformed into objectives within non-linear optimization problems and which algorithms can be used to solve them

Vector-Based 3D Graphic Statics: Transformations of Force Diagrams

The reciprocity between form and force diagrams in 2D graphic statics makes it possible to manipulate the form diagram while directly evaluating the redistribution of the forces within the force diagram. Conversely, after modifying the force diagram, the consequent transformation of the form diagram can be assessed at once. In the case of vector-based 3D graphic statics, the reciprocity between the diagrams is generally not achieved. This paper describes a series of transformations that can be applied to a vector-based 3D force diagram while allowing the corresponding 3D form diagram to adjust accordingly. Two categories of manipulations of the force diagram are described: global transformations that affect simultaneously all the elements of the diagram and local transformations, which permit the manipulation of individual elements of the diagram. Thanks to these transformations, the adjustment of the magnitude and the direction of the forces in vector-based 3D force diagrams can be used as an active operation in the structural design process

Combinatorial Equilibrium Modelling: A computational framework for equilibrium-based structural design

This dissertation presents the Combinatorial Equilibrium Modelling (CEM), a new structural design framework for the early conceptual design phase at the interface between architecture and engineering. The developed framework supports the interactive generation, evaluation and transformation of unconventional mixed (compression-tension) equilibrium models beyond funicular systems. Grounded on the lower bound theorem of the theory of plasticity, the proposed framework is focused on the diagrammatic visualization of equilibrium states. These diagrammatic models are helpful to enhance the dialogue between essential architectural and structural aspects in the early design phase, in both a solution-oriented (explorative) mode, and in the problem-oriented (convergent) mode. In the explorative design phase, designers usually search for satisfying solutions by alternately generating and evaluating design proposals, rather than trying to understand the nature of the ill-defined design problem. One can argue generating radically different proposals in this early design stage would help to obtain a clearer picture of the essential features of the design problem. In the field of structural design, however, only a limited number of proposals that are directly derived from a catalogue of well-known structural typologies are commonly considered. In this regard, this dissertation promotes an alternative strategy, which is characterized by going back to fundamental principles and originating design proposals through combining basic structural considerations. At the core of the proposed CEM framework, the CEM algorithm is able to generate at speed equilibrium models for any given combination of compression and tension edges and the applied loads. The CEM algorithm is based on a sequential partitioning and consequent generation of the equilibrium model. The algorithm relies on a topological diagram on top of the metric form and force diagrams known from graphic statics. This topological diagram supports a visual-oriented control of the qualitative load-bearing behaviour (i.e. connectivity and combination of inner force states) of discrete equilibrium models beyond conventional typological bounds. The CEM framework covers the initialization, the evaluation and the transformation of these equilibrium models in the light of the topological as well as the metric dimensions. In the transformation step, existing optimization algorithms can be additionally utilized to address constraints in both the form and the force domains. Design experiments highlight that these operations can be executed in many different ways and that the CEM framework can be used in both a bottom-up and in a top-down design mode. The real-time applicability of the developed framework is enabled by a computational implementation of the CEM algorithm into a digital toolkit. Depending on the context, the generated equilibrium models can be the ideal base for the conception of form-active, vector-active or section-active structures. The potential of the proposed framework is demonstrated through a series of case studies for bridges, multi-story buildings and stadiums

Conceptual design of three-dimensional structural and sun-shading façades supported by machine-learning

Façade design is a multi-disciplinary process that involves qualitative and quantitative aspects belonging to diverse disciplines – such as architecture, structural design and building physics. Since decisions made in the early design stage generally have a strong influence on the design outcome, the need for a holistic design strategy is evident. This paper presents an extension of a previous research work carried out by the authors regarding a holistic approach to the conceptual design of load-bearing and sun-shading façade systems. In this work, design aspects related to different disciplines are integrated into a holistic process thanks to the application of geometry-based methods (like graphic statics and solar geometry). At the same time, high-dimensional solution spaces are managed with the help of Self-Organizing Maps (SOMs). Particularly, the current paper focuses on the possibility to work with variable topologies. The proposed approach is applied in design explorations based on a multi-story building designed by the architect Christian Kerez and the structural engineer Joseph Schwartz. The case study shows how geometrical descriptions of the core design aspects supported by machine learning clustering algorithms support the designer in performing comprehensive design explorations and in making more conscious design choices

Holistic Design Explorations of Building Envelopes Supported by Machine Learning

The design of building envelopes requires a negotiation between qualitative and quantitative aspects belonging to different disciplines, such as architecture, structural design, and building physics. In contrast to hierarchical linear approaches in which various design aspects are considered and conceived sequentially, holistic frameworks allow such aspects to be taken into consideration simultaneously. However, these multi-disciplinary approaches often lead to the formulation of complex high-dimensional design spaces of solutions that are generally not easy to handle manually. Computational optimisation techniques may offer a solution to this problem; however, they mainly focus on quantitative aspects, not always guaranteeing the flexibility and interactive responsiveness designers need in the early design stage. The use of intuitive geometry-based generative tools, in combination with machine learning algorithms, is a way to overcome the issues that arise when dealing with multi-dimensional design spaces without necessarily replacing the designer with the machine. The presented research follows a human-centred design framework in which the machine assists the human designer in generating, evaluating, and clustering large sets of design options. Through a case study, this paper suggests ways of making use of interactive tools that do not overlook the performance criteria or personal preferences of the designer while preserving the simplicity and flexibility needed in the early design stage

Data-Driven Design: Exploring new Structural Forms using Machine Learning and Graphic Statics

The aim of this research is to introduce a novel structural design process that allows architects and engineers to extend their typical design space horizon and thereby promoting the idea of creativity in structural design. The theoretical base of this work builds on the combination of structural form-finding and state-of-the-art machine learning algorithms. In the first step of the process, Combinatorial Equilibrium Modelling (CEM) is used to generate a large variety of spatial networks in equilibrium for given input parameters. In the second step, these networks are clustered and represented in a form-map through the implementation of a Self Organizing Map (SOM) algorithm. In the third step, the solution space is interpreted with the help of a Uniform Manifold Approximation and Projection algorithm (UMAP). This allows gaining important insights in the structure of the solution space. A specific case study is used to illustrate how the infinite equilibrium states of a given topology can be defined and represented by clusters. Furthermore, three classes, related to the non-linear interaction between the input parameters and the form space, are verified and a statement about the entire manifold of the solution space of the case study is made. To conclude, this work presents an innovative approach on how the manifold of a solution space can be grasped with a minimum amount of data and how to operate within the manifold in order to increase the diversity of solutions.ISSN:2518-658

Vector-Based 3D Graphic Statics (Part I): Evaluation of Global Equilibrium

The evaluation of the equilibrium of a system of forces that fulfils specified boundary conditions is a core question of theory of structures. This paper reviews three methods, related to procedures introduced at the end of the 19th century, to evaluate the global equilibrium in three dimensions using graphic statics. The paper is specifically focused on one of these methods, which is grounded on the use of projections. Based on this method, a given system of forces can be reduced to three skew resultants, which are parallel to three initially chosen unit vectors. The three resultants can be composed into two resultants thanks to the construction of a simple 3D auxiliary structure or reduced to one resultant and a couple. Given the three resultants, the reactions at the supports can be evaluated according to specified boundary conditions in both cases of externally statically determinate and indeterminate systems

The Canopy: A Lightweight Spatial Installation Informed by Graphic Statics

This paper illustrates the design and fabrication process of the temporary installation The Canopy, developed as part of the fib Symposium on Conceptual Design of Structures 2021. The geometry of the perforated hanging membrane that forms The Canopy is the result of seamless integration between the disciplines of architecture and structural design, which was one of the driving inputs for the entire process. Particularly, the use of geometry-based models and graphic statics allowed activating the interplay between these disciplines. This was the key to balancing the relationship between architectural spaces and structural requirements, and to informing the multifaceted design exploration of The Canopy from conceptual design to construction.ISSN:2075-530

Vector-Based 3D graphic statics (Part III): Designing with Combinatorial Equilibrium Modelling

This paper presents an extension of the graphic-statics-based framework called Combinatorial Equilibrium Modelling (CEM). CEM allows for the generation of topologically and combinatorially different spatial equilibrium solutions in the early explorative design phase. In addition to the form and the force diagrams, CEM introduces a topological diagram, which enables the possibility to easily adjust the connectivity of the structure, the combinatorial state (tension-compression) as well as the static action (product of each element’s length and its corresponding force magnitude) of its inner forces. Thanks to its planarity, the topological diagram is always visually readable, comprehensible and easy to control, even in case of complex spatial structures. This innovative approach has the potential to find novel spatial networks in which the intrinsic structural properties can be controlled by the designer with simple visual operations

Stock-constrained truss design exploration through combinatorial equilibrium modeling

Reusing structural components has potential to reduce environmental impacts of building structures because it reduces new material use, energy consumption, and waste. When designing structures through reuse, available element characteristics become a design input. This paper presents a new computational workflow to design structures made of reused and new elements. The workflow combines Combinatorial Equilibrium Modeling, efficient Best-Fit heuristics, and Life Cycle Assessment to explore different design options in a user-interactive way and with almost real-time feedback. The method applicability is demonstrated by a realistic case study. Results show that structures combining reused and new elements have a significantly lower environmental impact than solutions made of new material only.SX