Surface-based inflatables are composed of two thin layers of nearly inextensible sheet material joined together along carefully selected fusing curves. During inflation, pressure forces separate the two sheets to maximize the enclosed volume. The fusing curves restrict this expansion, leading to a spatially varying in-plane contraction and hence metric frustration. The inflated structure settles into a 3D equilibrium that balances pressure forces with the internal elastic forces of the sheets.
We present a computational framework for analyzing and designing surface-based inflatable structures with arbitrary fusing patterns. Our approach employs numerical homogenization to characterize the behavior of parametric families of periodic inflatable patch geometries, which can then be combined to tessellate the sheet with smoothly varying patterns. We propose a novel parametrization of the underlying deformation space that allows accurate, efficient, and systematical analysis of the stretching and bending behavior of inflated patches with potentially open boundaries.
We apply our homogenization algorithm to create a database of geometrically diverse fusing patterns spanning a wide range of material properties and deformation characteristics. This database is employed in an inverse design algorithm that solves for fusing curves to best approximate a given input target surface. Local patches are selected and blended to form a global network of curves based on a geometric flattening algorithm. These fusing curves are then further optimized to minimize the distance of the deployed structure to target surface. We show that this approach offers greater flexibility to approximate given target geometries compared to previous work while significantly improving structural performance.
To compute the mechanical properties of different fusing curve patterns, we apply numerical homogenization and simulate unit cells of the fusing patterns under periodic boundary condition
Physics-based simulation of the deployment of several surface-based inflatables
We employ a two-scale optimization approach. If the coarse-scale optimization does not fully reproduce the target shape, we run a nested optimization with simulation in the loop to fine-tune the fusing curve geometry
We built physical prototypes to validate our design algorithms. Here we show the deployment process of two surface-based inflatables with the same target shape, but the first one is optimized using the parallel tube pattern while the second one is optimized using the Cosine Curve pattern family which covers a much larger range of mechanical properties. We see that the second structure approximate the target shape much closer and has better structural stability
Deployment process of other physical prototypes. The video is played at 3x speed
Surface-based inflatable unit cell. Two thin, planar sheets are joined together along a pattern of fusing curves (top right). Certain regions can be cut out to create a permeable cell (bottom left). Using periodic boundary conditions, we can simulate inflation, which causes the cell to bulge verti- cally and contract in plane (bottom right). Our framework characterizes the geometric and physical properties of unit cells with arbitrary fusing and cutting patterns.
Overview of our computational pipeline. Using a novel numerical homogenization approach, we process a large set of parameterized unit cell patches to evaluate their in-plane contraction as well as stretching and bending stiffnesses. Our inverse design optimization tracks the inflated equilibrium state and blends suitably chosen unit cells with good structural performance to best approximate a given target surface.
Tiling a unit cell.The model in the bottom left has been simulated with periodic boundary conditions as used in homogenization. The simulated and fabricated tilings with closed boundaries indicate how the geometric and physical properties of the center patch of these grids converge to the periodic result as the grid extends.
Representative unit cells for the different pattern families. From top to bottom: fusing curves, inflated patch, bending stiffness profile, and stretching stiffness profile
Properties of unit cell pattern families. From left to right: in-plane reciprocal stretch factors in principal directions, minimal and maximal bending stiffness, and minimal and maximal stretching stiffness. Colors represent minimal bending stiffness for each patch, where darker colors indicate higher values.
Impact on the inflated shape of a fictitious gravity force. On the left, no external forces are applied; following the arrow, the gravity force increases. The simulation indicates a complete collapse and inversion of the parallel tube model while the cosine pattern model maintains its shape well.
Computational Homogenization for Coarsened Design Optimization. For a given target surface and pattern family, the local/global flattening computes a first approximation of the scale factor distribution and corresponding cell orientation. These are converted to pattern parameters by the subsequent optimization, which also significantly improves the target fitting and globally adjusts the fusing curves pattern. Color-coding indicates the distance to the target surface relative to the bounding box diagonal.
Fusing curves (right) are generated by merging together per-triangle polylines defined by the pattern parameter field $\p$ and a high-frequency stripe pattern texture map.
This model, optimized with the elliptical holes pattern, illustrates the importance of the full-scale shape optimization in the final stage of our inverse design pipeline
Fabrication setup (left) and material failures (right). Damage in the fabric can occur due to melting that leads to ruptures when over-inflating.
Comparison of our homogenization results with the zigzag-pattern- specific geometric abstraction proposed in [Gao et al. 2020]. The ticks on each vertical line in the bending stiffness distribution plot indicate maximum, minimum, and mean values. Unit cell patterns generated by varying 𝛼 are shown in the bottom row.
Comparison of parallel tubes (left) with our approach (right). The weak bending stiffness across the tube directions causes the free boundary roof to collapse into a sharp kink, while our approach avoids this weakness and reproduces the target shape more faithfully. Color-coding indicates distance to the target surface (overlayed with transparency) relative to the bounding box diagonal.
Comparisons between simulation and physical prototypes for the parallel tubes of [Panetta et al. 2021] and our approach. The bottom row shows a basic load-testing experiment. See also accompanying video for more loading scenarios.
Using the elliptical holes pattern family, we construct a permeable neck brace that can stabilize an injured patient’s head, illustrating a poten- tial medical application. The fabricated sheet has been re-connected along the congruent boundary seam lines prior to inflation.
The dash line pattern used on a surface with positive and negative Gaussian curvature. łTeardropsž have been added to the open ends of the line segments to reduce the risk of decollation at higher inflation pressure.
Deviation of physical models from input surfaces. We scaned the fabricated models and color-coded the results using the distance to the closest point on the design surface relative to the bounding box diagonal.
A simple parametric family of unit cells having controllable non- planar equilibrium states.
Surface-based inflatable unit cell. Two thin, planar sheets are joined together along a pattern of fusing curves (top right). Certain regions can be cut out to create a permeable cell (bottom left). Using periodic boundary conditions, we can simulate inflation, which causes the cell to bulge verti- cally and contract in plane (bottom right). Our framework characterizes the geometric and physical properties of unit cells with arbitrary fusing and cutting patterns.Overview of our computational pipeline. Using a novel numerical homogenization approach, we process a large set of parameterized unit cell patches to evaluate their in-plane contraction as well as stretching and bending stiffnesses. Our inverse design optimization tracks the inflated equilibrium state and blends suitably chosen unit cells with good structural performance to best approximate a given target surface.Tiling a unit cell.The model in the bottom left has been simulated with periodic boundary conditions as used in homogenization. The simulated and fabricated tilings with closed boundaries indicate how the geometric and physical properties of the center patch of these grids converge to the periodic result as the grid extends. Representative unit cells for the different pattern families. From top to bottom: fusing curves, inflated patch, bending stiffness profile, and stretching stiffness profileThefivebasisfunctionsforsingle-axisbendingstiffness𝑘𝑏(𝛼).Selected unit cells of the Random Voronoi familyProperties of unit cell pattern families. From left to right: in-plane reciprocal stretch factors in principal directions, minimal and maximal bending stiffness, and minimal and maximal stretching stiffness. Colors represent minimal bending stiffness for each patch, where darker colors indicate higher values.Impact on the inflated shape of a fictitious gravity force. On the left, no external forces are applied; following the arrow, the gravity force increases. The simulation indicates a complete collapse and inversion of the parallel tube model while the cosine pattern model maintains its shape well.Computational Homogenization for Coarsened Design Optimization. For a given target surface and pattern family, the local/global flattening computes a first approximation of the scale factor distribution and corresponding cell orientation. These are converted to pattern parameters by the subsequent optimization, which also significantly improves the target fitting and globally adjusts the fusing curves pattern. Color-coding indicates the distance to the target surface relative to the bounding box diagonal.Fusing curves (right) are generated by merging together per-triangle polylines defined by the pattern parameter field $\p$ and a high-frequency stripe pattern texture map. This model, optimized with the elliptical holes pattern, illustrates the importance of the full-scale shape optimization in the final stage of our inverse design pipeline Fabrication setup (left) and material failures (right). Damage in the fabric can occur due to melting that leads to ruptures when over-inflating. Comparison of our homogenization results with the zigzag-pattern- specific geometric abstraction proposed in [Gao et al. 2020]. The ticks on each vertical line in the bending stiffness distribution plot indicate maximum, minimum, and mean values. Unit cell patterns generated by varying 𝛼 are shown in the bottom row. Comparison of parallel tubes (left) with our approach (right). The weak bending stiffness across the tube directions causes the free boundary roof to collapse into a sharp kink, while our approach avoids this weakness and reproduces the target shape more faithfully. Color-coding indicates distance to the target surface (overlayed with transparency) relative to the bounding box diagonal. Comparisons between simulation and physical prototypes for the parallel tubes of [Panetta et al. 2021] and our approach. The bottom row shows a basic load-testing experiment. See also accompanying video for more loading scenarios. Using the elliptical holes pattern family, we construct a permeable neck brace that can stabilize an injured patient’s head, illustrating a poten- tial medical application. The fabricated sheet has been re-connected along the congruent boundary seam lines prior to inflation. The dash line pattern used on a surface with positive and negative Gaussian curvature. łTeardropsž have been added to the open ends of the line segments to reduce the risk of decollation at higher inflation pressure. Deviation of physical models from input surfaces. We scaned the fabricated models and color-coded the results using the distance to the closest point on the design surface relative to the bounding box diagonal. A simple parametric family of unit cells having controllable non- planar equilibrium states.
Computational Homogenization for Inverse Design of Surface-based Inflatables
This work is built upon the inverse design framework introduced in a previous project from GCM, where the design algorithms focus on networks of parallel tubes. See the SIGGRAPH 2021 talk below for more details!
