Reconfigurable mold system and method for manufacturing curved panels

Abstract

A method (500) for making a doubly-curved panel (402) includes receiving (502) a target curvature profile of the doubly-curved panel (402), determining (504) boundary support curves (218) for a flexible membrane (110) for mimicking the target curvature profile, wherein the boundary support curves (218) define a surface that is larger than a surface defined by the target curvature profile, placing (506) a support frame (214) inside a housing (202), wherein a top perimeter of the support frame (214) follows the boundary support curves (218), attaching (508) the flexible membrane (110) to the housing (202), applying (510) a deflation pressure within the housing (202) so that the flexible membrane (110) deflates according to the boundary support curves (218) to form a mold that mimicks the doubly-curved panel (402), and forming (512) a material over the flexible membrane (110) to form the doubly-curved panel (402).

Description

RECONFIGURABLE MOLD SYSTEM AND METHOD FOR MANUFACTURING CURVED PANELSCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 710,781, filed on October 23, 2024, entitled “RECONFIGURABLE MOLD SYSTEM FOR THE PRODUCTION OF THIN DOUBLY-CURVED PANELS AND SIMILAR APPLICATIONS,” the disclosure of which is incorporated herein by reference in its entirety.BACKGROUND OF THE INVENTION TECHNICAL FIELD

[0002] Embodiments of the subject matter disclosed herein generally relate to methods and reconfigurable mold systems for manufacturing curved objects, for example, panels having complex, free-form, or doubly-curved surfaces.DISCUSSION OF THE BACKGROUND

[0003] Panel manufacturers encounter significant challenges when they produce doubly-curved panels for architectural facades. Typically, they create a custom mold for each unique panel shape by machining a large block of material, such as polystyrene or polyurethane foam, using computerized numerical control(CNC) methods. After casting or forming the panel, the panel manufacturers usually discard the mold. This approach consumes substantial resources, generates considerable waste, and requires significant time and expense, especially when projects require many unique panels. The cost and waste of single-use molds restrict the economic viability of free-form designs to high-profile projects.

[0004] Various research teams have introduced reconfigurable mold systems to address cost and waste issues. One category of these systems employs a dense grid of computer-controlled pins or actuators, whose tips collectively define the desired mold surface. In certain configurations, the operator places elastic material over the pin grid to achieve a smoother surface finish. However, pin-based systems exhibit drawbacks: they require a large number of actuators and complex mechanical and control components, which increase both cost and complexity. Additionally, the resolution and smoothness of the mold surface depend on actuator density, and discrete actuator points may leave visible artifacts on the finished panel.

[0005] Some researchers have explored flexible molding using membranes, but existing methods possess notable limitations. For example, some systems use a membrane without applying pressure, which severely restricts the range of achievable shapes, especially those with positive Gaussian curvature. Other systems may not include a computational framework for solving the inverse design problem, i.e., determining the precise boundary conditions and system parameters needed to form a specific target shape. Without a reliable solution to this inverse problem, the panel manufacturers struggle to accurately reproduce arbitrary panelgeometries. Additionally, these methods may lack mechanisms for pre-stretching the membrane, which further limits the range of shapes that can be fabricated.

[0006] Architectural designers have implemented pneumatic systems in various forms, often on a larger scale rather than for high-precision panel fabrication. Inflating large structures to approximate shapes is less demanding than achieving precise control for smaller components. Examples include pneumatic formwork for concrete shells [Kromoser and Kollegger 2015] and the erection of gridshells [Quinn and Gengnagel 2014], Simulation-based design presents a challenge in this field, and a recent contribution includes an inverse design solution for surface-based inflatables [Panetta et al. 2021],

[0007] By employing a pneumatic system, designers aim to develop a cost-effective and sustainable method for fabricating architectural panels. To reduce costs in freeform architecture, practitioners often use rationalization, which involves approximating a design with a simpler, more economical one [Fischer 2012;Pottmann et al. 2015]. For example, Eigensatz et al.

[2010] presented a paneling solution that reduces the number of custom molds required, though this approach may introduce visible kinks between panels.

[0008] Paneling techniques depend on the design and material properties. For panels isometric or almost isometric to the plane sheet metal or other bendable materials can be bent in situ without any mold. For the fabrication of developable panels from other materials bendable materials can be used to efficiently build molds. Even brittle materials such as glass can be bent into curved shapes if the shapes are carefully designed with a fabrication-aware design system.

[0009] Pellis et al. [2021; 2020] rationalize individual surfaces using corresponding Weingarten surfaces, which are characterized by a relationship between their principal curvatures and hold significance in differential geometry. Rogers and Schief

[2003] identified their connection to equilibrium shapes of membranes under normal loading. However, the assumption about curvature-stress relations does not suit real-world membranes with nonlinear behavior.

[0010] Panel manufacturers can achieve complex panels by using a thin shell model. Scientific communities, including those focused on mathematical modeling, computational physics, and computer engineering, have addressed the thin shell model using thin shell simulations. Using a thin shell model instead of a classical finite element method (FEM) offers several advantages. The thin shell model operates more efficiently because it uses fewer elements to represent the membrane and accurately considers the membrane's bending behavior and thickness. This efficiency allows for faster analysis and simulation of the membrane’s behavior under various loading conditions and demonstrates capabilities in predicting the membrane’s response to external loads and deformations. Furthermore, a thin shell model provides a simplified, intuitive representation of the membrane, which helps users understand and interpret results.

[0011] The visual computing community has contributed sophisticated methods for accurate and efficient simulation of thin shells and related problems, such as handling collisions and interpenetrations. Researchers still investigate related issues, including simulating cloth dynamics, folding and crumpling sheets, and the behavior of fiber meshes. Because numerically handling the underlyingdifferential equations remains challenging, experts have devised a diverse portfolio of integration methods for elastodynamic systems [Chen et al. 2020; Michels et al. 2017, 2014],

[0012] While current solutions address some freeform fabrication challenges, they suffer from significant drawbacks. Conventional methods that rely on single-use, CNC-machined molds consume resources, generate waste, and incur high costs, limiting their use to high-profile projects. Pin-based reconfigurable mold systems offer more flexibility but remain mechanically complex and expensive, with fabrication resolution constrained by actuator density, which can leave undesirable artifacts on panel surfaces. Approaches that use membranes often struggle to produce geometrically complex shapes, especially those with positive Gaussian curvature or can only accurately reproduce the panel boundaries, not the entire shape of the target panel. Furthermore, many existing systems lack a robust computational framework for solving the inverse design problem (i.e., determining the precise boundary conditions needed to form a specific target shape), which makes it difficult to reliably and accurately fabricate arbitrary panel geometries.

[0013] The industry needs a reconfigurable mold system that reduces complexity and cost compared to pin-based systems, while forming a wide range of complex shapes with high precision. An effective system needs to include a robust computational framework for solving the inverse design problem, so users can accurately determine the required boundary conditions for any given target panel shape.SUMMARY OF THE INVENTION

[0014] According to an embodiment, there is a method for making a doubly-curved panel, and the method includes receiving a target curvature profile of the doubly-curved panel, determining boundary support curves for a flexible membrane for mimicking the target curvature profile, wherein the boundary support curves define a surface that is larger than a surface defined by the target curvature profile, placing a support frame inside a housing, wherein a top perimeter of the support frame follows the boundary support curves, attaching the flexible membrane to the housing, applying a deflation pressure within the housing so that the flexible membrane deflates according to the boundary support curves to form a mold that mimicks the doubly-curved panel, and forming a material over the flexible membrane to form the doubly-curved panel.

[0015] According to another embodiment, there is a reconfigurable mold system for making a doubly-curved panel, and the system includes a housing, a support frame disposed within the housing, wherein the support frame has a top perimeter that follows boundary support curves that define a surface that is larger than a surface of a target curvature profile of the doubly-curved panel, wherein the target curvature profile represents a non-standard, three-dimensional, free-form geometry or shape of the doubly-curved panel, a flexible membrane that is attached to the housing so that the flexible membrane deflates according to the boundary support curves, and a first vacuum pump fluidly connected to the housing to draw air away from the housing to deflate the flexible membrane.BRIEF DESCRIPTION OF THE DRAWINGS

[0016] For a more complete understanding of the present invention, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

[0017] FIG. 1 illustrates a flow chart for modeling a target panel shape and reproducing that shape using a flexible membrane in a reconfigurable mold system.

[0018] FIG. 2 presents a schematic diagram of a reconfigurable mold system that uses a flexible membrane to dynamically achieve a desired target panel shape.

[0019] FIG. 3 provides an overview of the reconfigurable mold system, showing a support frame that holds the flexible membrane.

[0020] FIG. 4A schematically depicts the curved flexible membrane, the curved panel to be fabricated on it, and various parameters associated with these elements.

[0021] FIG. 4B schematically shows an extended Weingarten surface approximating the curved target panel.

[0022] FIG. 5 displays a flow chart of a method for manufacturing a target curved panel using a flexible membrane in the reconfigurable mold system shown in FIG. 2.

[0023] FIG. 6 illustrates various curved panels created with the method described in FIG. 5 and the reconfigurable mold system of FIG. 2.

[0024] FIG. 7 presents a table that lists parameters for the curved panels shown in FIG. 6.

[0025] FIG. 8 shows a schematic diagram of a computing device that may host one or more of the methods discussed in this document.DETAILED DESCRIPTION OF THE INVENTION

[0026] The following description of the embodiments refers to the accompanying drawings. The same reference numbers in different drawings identify the same or similar elements. The following detailed description does not limit the invention. Instead, the scope of the invention is defined by the appended claims. The following embodiments are discussed, for simplicity, with regard to a flexible membrane which is shaped relative to a support frame that has four side panels. However, the embodiments to be discussed next are not limited to a support frame having four side panels but may be applied to support frames that have 3 or 5 or more side panels. A support frame having four planar curves are just one way to create specific boundary conditions for the membrane. Other methods include using any number of non-planar curves, planar curves, points, or any combination of these. In one aspect, the molding is independent of the specific object being produced or the method used. The flexible membrane surface may replace whatever mold would have been used previously.

[0027] Each occurrence of “one embodiment” or “an embodiment” signals the presence of a specific feature, structure, or characteristic in at least one embodiment of the disclosed subject matter. The phrases “in one embodiment” or “in an embodiment” in multiple locations do not necessarily indicate the same embodiment. Suitable combinations of features, structures, or characteristics may occur across one or more embodiments.

[0028] One embodiment presents a novel reconfigurable mold system and associated controller for producing a curved object, e.g., doubly-curved panels. The system uses a flexible membrane as a reconfigurable mold surface. To generate a curved panel with a specified input shape (the target panel), the system is configured to calculate the required boundary condition and deflation pressure to apply to the flexible membrane. More details about this system and associated method are now discussed.

[0029] Panel manufacturing typically involves casting a material onto a base, termed a mold, to achieve the desired panel geometry. FIG. 1 offers a high-level overview of a method 100 for obtaining such a curved panel. The process begins with a reference panel 101 derived from a design surface 102, typically received from a client (step 103). The reference panel 101 undergoes a mathematical approximation (step 105) by a quadratic Weingarten surface 104, which yields an estimated set of support curves 106. This estimation initializes an inverse simulation loop (step 109) to optimize the support curves 106. The system then reconfigures a mold 108 according to the optimized support curves 106 (step 111), followed by a deflation of a flexible membrane 110 to serve as mold 108 (step 113).

[0030] Traditional panel manufacturing relies on static molds. The design of a flexible mold requires a resilient yet flexible surface capable of withstanding the panel’s weight while enabling reconfiguration into various shapes to meet production requirements. One embodiment identifies a suitable flexible material for the panel-supporting surface and engineers a mechanism to provide sufficient support force. One embodiment implements a pressure force of constant magnitude to deflate adeformable membrane beneath which a support frame is positioned, thus forming the flexible mold.

[0031] FIG. 2 schematically depicts a reconfigurable mold system 200, which supports the flexible (deformable) membrane 110. The flexible membrane 110 may have any shape, e.g., circular, rectangular, triangular, etc. System 200 includes a housing 202, constructed to support the flexible membrane 110, a panel (not shown) to be formed atop the flexible membrane 110, and to withstand a negative pressure inside a chamber 203. A first vacuum pump 204, located inside or outside housing 202, generates the negative pressure P1 and connects to the chamber 203 via piping 206. While the embodiment shown in the figure uses two pumps, one skilled in the art would understand that another embodiment may use a single pump. Also, one skilled in the art would understand that other type of pumps (e.g., an electric pump) may be used with similar results. A controller 208 (for example, a processor) regulates the pressure generated by the first pump 204. The housing 202 may be modular, comprising multiple units 210 that stack to increase a height H as required by the panel specifications.

[0032] FIG. 2 also shows a support frame 214, placed inside the chamber 203 of the housing 202 to support the flexible membrane 110. The support frame 214 is made of four panels (planes) 2161, indexed by I = 4. Only panels 216-1 to 216-3 are visible in the cross-section of the reconfigurable mold system 200. The support frame 214 is tailored for each panel shape; panels with different shapes require distinct support frames, as the top edge geometry of the panels forming the support frame varies. The base of each panel 216I is flat to fit the bottom of the housing 202,and the top edge of each panel 2161 is curved to define the target panel’s geometry. The top edge of each panel 2161 may possess a unique curve. The four top edges of the panels 2161 establish the boundary support curves 218 for the flexible membrane 110, as depicted in FIG. 3. The top edge of each panel 2161, for example panel 216-3 in FIG. 2, may feature a bullnose profile to increase friction with the flexible membrane 110 and prevent membrane slippage during deflation. The rounded edge prevents damaging the membrane, and a slightly rough texture ensures a no-slip contact. Any soft, round shape may work, but using a circular pipe surface has some computational advantages. More specifically, to ensure effective panel formation, the flexible membrane 110 needs to reliably transmit the applied pressure and conform precisely to the boundary support curves defined by the support frame 214. The interaction between the flexible membrane 110 and the bullnose profiles of the support panels 216I helps maintain positional accuracy and prevents unwanted displacement during the molding process. Furthermore, the modular design of housing 202 allows for easy adaptation to panels of varying dimensions, enhancing the versatility of the mold system for different architectural or industrial applications.

[0033] One embodiment incorporates a top edge or frame 220, fastened to the top of the housing 202 to retain the edges of flexible membrane 110 in direct contact with the housing 202. The housing 202 remains open at the top when the flexible membrane 110 is absent. The top edge 220 secures the flexible membrane 110 against the rim of the housing 202, preventing air escape during the deflation process. A required prestress in the membrane 110 may be created by means otherthan raising the support frame 214, for example, by pushing on or pulling the flexible membrane outside the top edge 220.

[0034] FIG. 2 also shows a distance sensor 224 placed inside the chamber 203 and connected to the controller 208, to measure the flexible membrane 110 deformation. In one embodiment, the distance sensor is replaced with a laser scanner to capture a 3D shape of the relevant part of the membrane and use the full 3D information to control the pressure, and, if present, the positions of the actuators forming the articulated or adjustable support structure. This 3D digitization of the relevant flexible membrane area occurs from within the housing 202. In one embodiment, this data is used not only to adjust the pins but also for monitoring and quality control. For instance, it allows for diagnosing the flexible membrane's condition to indicate when it needs to be replaced. This data may also be added to the documentation for each panel. A variation of this embodiment includes an additional housing 230 mounted atop housing 202, with a second flexible membrane 232 attached to the top of additional housing 230, forming a second chamber 234 between the first flexible membrane 110 and the second flexible membrane 232. The second flexible membrane 232 may be made of the same or different materials as the first flexible membrane 110. A second pump 236, connected via piping 238 to the second chamber 234, applies a second pressure P2, distinct from the first pressure P1. The controller 208 also operates the second pump 236. The second membrane 232 presses the material cast over the first flexible membrane 110 to achieve the desired panel shape and to increase the density of the manufactured panel and / or a top smoothness of the manufactured panel. The first flexible membrane 110 ensuresa bottom smoothness of the manufactured panel. In one embodiment, both pressures may be larger or smaller than the surrounding air pressure. In yet another embodiment, instead of air, non-compressible liquids may be used.

[0035] In one embodiment, the reconfigurable mold system 200 may include one or more adjustable pin mechanisms 240 for further shaping the flexible membrane 110. The pin mechanisms 240 may include one or more support rods or similar mechanisms to bridge the gaps between the pins and create a smooth support for contacting the membrane. For simplicity, FIG. 2 shows only one adjustable pin mechanism 240. However, many such mechanisms may be used. FIG. 2 shows the mechanism 240 placed inside the chamber 203. In one embodiment, the mechanism 240 is placed outside the chamber 203, to impinge on the flexible membrane 110 from above. In yet another embodiment, it is possible to have a mixture of inside and outside mechanisms 240. The mechanism 240 includes an actuator 242 (e.g., a motor) and a pin 244. The actuator 242 is configured to move the pin 244 up and down as indicated by the controller 208. By moving the pin 244 against the flexible membrane 110, the surface of the flexible membrane may be further adjusted. In one embodiment, flexible members (not shown) may connect the top of two or more pins for impinging on the surface of the flexible membrane.

[0036] For the reconfigurable mold system 200, the inventors have configured the controller 208 to solve an inversion problem to determine the boundary conditions and system parameters needed to form a specific target panel shape using the flexible membrane 110. This means that the controller 208, among other things, can calculate the boundary support curves 218 so that the support frame 214can be manufactured before deforming the flexible membrane 110. If the pin mechanisms discussed above are used, the controller may be configured to compute the positions of the pins, move the pins to the desired positions, and later perform fine adjustments based on the actual 3D shape of the membrane. As discussed later, there is a gap between the boundary support curves 218 and the panel to be fabricated. This gap increases the degrees of freedom and allows to approximate not only the target panel boundaries but the panel shape.

[0037] The mathematical modeling of the flexible membrane 110 employs the Kirchhoff- Love assumption, representing membrane as fl x [~h / 2,h / 2] c ]R3, where fl denotes the rest or undeformed midsurface of the membrane and h denotes the membrane thickness. A deformation map function <p:fl -> fl maps the undeformed midsurface fl into a deformed midsurface fl, enabling the membrane 5 to assume various shapes. The deformed midsurface fl may be reconfigured to match a target panel T c K3. All references to the membrane in this document refer to the deformed midsurface fl of 5.

[0038] The support frame 214 placed beneath the membrane fl facilitates the formation of the ideal shape matching target panel T. During the fabrication process, the flexible membrane 110 deforms to match the shape of the target panel T. After producing each panel, the flexible membrane 110 restores its original shape by equalizing the pressure with the ambient levels (p — > 0), imparting the term “reconfigurable mold” to the system 200. The flexibility results from both the elastic properties of the flexible membrane 110 and the replaceable support frame 214, enabling production of panels in various shapes.

[0039] A process for determining the boundary support curves 218 in the controller 208 proceeds as follows. The placement of the support frame 214 and the application of a constant pressure p inside the chamber 203 primarily determine the shape of the deformable membrane fl. The support planes 2161 of the support frame 214 directly contact the membrane fl, exclusively at top bullnose edges, allowing modeling of the support planes 216I as a set of four-piece pipe surfaces of radius r, with centerlines denoted by y = { / i, y2, y3, }- This four-piece pipe surface model suits support frame 214 with four support planes 2161; alternative configurations including 3 or more than 4 support planes may be used. In one embodiment, the support frame may include point-like supports, for example, mushroom shaped supports, or a combination of curves and pins.

[0040] Given a constant deflating pressure p inside the chamber 203, positioning of the pipe surface centerlines y determines the static equilibrium shape of the membrane fl and defines an integrated L2distance between the target panel T and the membrane fl. A formulation from Iterative Closest Points (ICP) for rigid registration [Pottmann et al. 2006] adapts the linearized integrated L2distance for the optimization of an objective function 2):r 72)(T,fl) = ((y -projfi(y))n) <±4 (1)Jyerwhere y denotes a point on the target panel T, proj_Q(y represents the closest-point projection operator of y onto the membrane fl, and n is the unit normal vector of the tangent plane at proj_Q(y e fl. Note that the ICP is an algorithm in computer vision and computational geometry for aligning two shapes, typically point clouds orsurfaces, in three-dimensional space. Its goal is to find the best rigid transformation (rotation and translation) that minimizes the distance between corresponding points of the two datasets. Metrics based on geometric measurements, such as Chamfer and Hausdorff distances, provide useful distance evaluations but lack suitability for optimization due to their non-differentiability. The distance definition in Equation (1) is termed the “point-to-plane” distance, a locally smoothed derivative of the Lm(Hausdorff) distance:fl) = max (sup inf ||y — x||,sup inf ||y — x|| (2)(yerxe-rix6nxe:r)where % is a point on the membrane ft.

[0041] While this distance definition remains geometric, the panel’s fabrication process requires the flexible membrane reaching a static equilibrium state, a physical process necessitating a physics-aware optimization algorithm to optimize placement of pipe surface centerlines y.

[0042] The process of modeling the target panel T treats the panel as a rigid body with a rigid pose (3?, t), where 3? e SO(3) c ]R3X3is a rotation matrix and t e K3is a translation vector. The rigid pose (3?, t) indicates that target panel T cannot bend or deform but may rotate by 3? and / or translate by t. The process proposed by the inventors to be used by the controller 208 employs a twofold optimization scheme: (1) a geometric initialization algorithm identifies the initial placement (shape and position) of the pipe surface centerlines y and the initial rigid pose (3?, t) of target panel T, and (2) a physics-aware optimization algorithm validates and enhances the design of the pipe surface centerlines y to optimize the shape of membrane Q forcloser alignment with the target panel T at the rigid pose (3?, t). In one embodiment, before the actual fabrication of the panel, the actual 3D shape of the membrane is captured (e.g., by a laser scanner) and a target shape is registered to this digitized surface, minimizing the error. The panel is then fabricated in this final position and orientation.

[0043] The geometric initialization process (1) and the simulation-based optimization of the pipe surface centerlines (2) proceed as follows. The housing 202 of the reconfigurable mold system 200 may match the undeformed membrane midsurface fl in width and length. Lightweight support planes 2161 may utilize medium density fiberboard (MDF) or other materials (e.g., plywood) for rapid prototyping. For each new panel, the fabrication of the four-piece support frame 214 occurs as required.

[0044] The panel fabrication process involves deflating the flexible membrane 110 by applying a negative constant pressure p inside the chamber 203. Note that it is possible, before deflating the membrane, to briefly inflate it until the membrane lifts completely from the support frame. This ensures that any uneven stress in the membrane introduced during the installation of the membrane (position of the pins) is released. In this way, the actual stress distribution matches at the start of the simulation. The flexible membrane 110 deflates under the action of the first pump 204 until reaching an equilibrium, where the forces are modeled by the static equilibrium equation:f (x) = fe(x) — ( / p(x) + mg) = 0, s.t. dy(x) > R = r + h / 2, (3) total internal externalwhere feis the membrane elastic force, fpis the pressure force, mg is the gravitational force, and the nonpenetration constraint dYdefines a minimum distance between the membrane Q and the pipe surface centerlines y, which needs to exceed the fillet pipe surface radius R, with membrane radius r and thickness h. FIG. 4A illustrates a panel 402, a Weingarten surface 404 approximating a target surface 406 of the panel 402, the radius R of a circle centered on the pipe surface centerline y, a distance d between an end of the panel 402 and a plane 408 parallel to the support plane 216I (and central to the support plane 216I), a flexible membrane 110 thickness h, and a bullnose edge radius r that characterizes the end of the support plane 216I. Note that the target surface 406 is an offset surface of the surface of the panel 402. In this application, the term “offset” refers to a process of moving each point of a surface (panel 402 in this case) perpendicularly away from the surface, by a certain distance. In this embodiment, the “certain distance” is half of the thickness h of the membrane 110. For the following calculations, the target surface 406 is used instead of the actual panel 402 surface 403.

[0045] The geometric initialization (process step (1)) with Weingarten surfaces proceeds as follows. The shape (i.e., a surface 403) of the panel 402 is known (provided by the company that needs the panels). The target surface 406 is determined by applying the offset operation noted above to the surface 403 of the panel 402. The target surface 406 may be approximated by a B-spline surface 405, which is larger than the target surface 406. The extended B-Spline surface 405 is used as the initialization for an optimization process. For example, the method may optimize control points of the B-Spline surface 405 so that (1) the surface is aWeingarten surface 404 (in other words fulfills a curvature relation 2AK + H + p = 0) and (2) the surface approximates the offset surface 406 of the target panel surface 403. In one embodiment, the Weingarten relation is prioritized over a closeness to the offset target panel shape. The B-Spline surface 405 is larger than the target panel 406 that is a Weingarten surface 404 or at least very close to a Weingarten surface. The Weingarten surface 404 is then offset with a distance R, which is given by the sum of (a) half a thickness h of the membrane 110, and (b) the radius r of the bullnose of the pipe. The offset of the Weingarten surface generates an offset surface 420, as shown schematically in FIGs. 4A and 4B. The offset surface 420 exceeds the area of the target surface 406 of the panel 402 by approximately 20% on each side; alternative values may range from 10% to 40%. In one embodiment, the B-spline surface 405 may be used to obtain the initial shape of the pipe surface center lines. These initial curves may also be found by directly evaluating the Weingarten surface 404 without using the B-spline surface 405. For this embodiment, the pipe surface center lines are obtained by intersecting the offset surface 420 with the plane 408, as schematically illustrated in FIG. 4A.

[0046] Thus, the Weingarten surface is used to model an ideal membrane that approximates the surface of the target panel. The Weingarten surface geometrically extends the target panel 402 surface 403 beyond its borders so that the simulation can capture how the flexible membrane 410 bends and stretches near the edges. It forms the foundation for extracting physically consistent boundary supports and initializing the optimization process for determining the mold shape.

[0047] The method extracts a four-piece curve set (the boundary support curves 218) from the intersection of the offset surface 420 with the plane 408, for the initial placement of the pipe surface centerlines y. During this step, the pose (3?, t) is defined relative to placement of y.

[0048] The initialization of the support curve shapes utilizes a geometric approach (step (1) discussed above) based on a simplified mechanical model of the flexible membrane 110 in the reconfigurable mold system 100, referencing Equation (3). The flexible membrane initially behaves as a shell membrane resisting only tensile stresses, loaded by constant pressure p in the direction of normal n to the membrane surface. A principal curvature parametrization of the deformed flexible membrane surface uses unit basis {e±, e2] and unit normal n = e1x e2. The principal curvatures Ki and K2along directions e±and e2correspond to membrane stresses N±and N2, respectively, with positive values indicating tension. Equilibrium in the normal direction requires the following relationship:N1K1+ N2K2+ p = 0. (4)

[0049] The following ansatz applies to the membrane stresses:N1= A\K^ + p, N2= \K2\ + p, (5) with A > 0 and p > 0 to ensure tension. This assumption arises from constraints of the flexible membrane in the reconfigurable mold system 100: higher curvature in a direction leads to greater stretching and correspondingly higher tensile stress.Equation (5) assumes a linear relationship between the absolute curvature and the tensile stress. A homogeneous pre-stretching of the flexible membrane via thecoefficient z is permitted. Various curvature and corresponding normal stress values appear in the simulations.

[0050] Substitution of Equation (5) into Equation (4) yields the following condition for the principal curvatures of the surface of the flexible membrane:K1|K2| + K2|K2I) + / *(«! + K2) + V = 0 with A > 0, z > 0. (6)

[0051] A surface for which the principal curvatures satisfy a functional relationship is classified as a Weingarten surface. Researchers have analyzed the connection between the equilibrium of shell membranes under normal loading and Weingarten surfaces, demonstrating that a shell membrane with principal stresses following the linear relations N±= K2+ z and N2= K±+ z achieves equilibrium under the constant pressure p when the shell adopts the shape of a Weingarten surface with the curvature relation 2 K + pH + p = 0. In these expressions, H denotes the mean curvature and K denotes the Gaussian curvature, given by H =1 / 2(KI + K2) and K = KPC2. For such surfaces, when the principal stresses align with the principal curvature directions e±, e2, equilibrium in the tangent plane of the membrane occurs automatically, as this configuration satisfies the Gauss-Mainardi-Codazzi equations of the surface. The specific geometric and mechanical characteristics of the pressurized flexible membrane 110 necessitated the development of an alternative geometric hypothesis on membrane stresses, as stated in Equation (5).

[0052] Equation (5) does not ensure tangent equilibrium. Nonetheless, the shell membrane model serves as an initialization (step (1) discussed above) for a more accurate thin shell simulation (step (2) mentioned above), making it sufficient to consider equilibrium in the normal direction only.

[0053] Considering the negative pressure p (a force of magnitude p acting opposite to the surface normal n), Equation (6) produces a curvature relation in the (KI, K2) plane, represented by hyperbolic branches in quadrants II and IV, with asymptotes along the bisecting line, and a circular segment in quadrant I, centered on its bisecting line. This curve remains entirely above the asymptotic line K±+ K2= 0. This property indicates that, with this flexible membrane model, the equilibrium under the negative pressure p occurs only for the surfaces exhibiting a positive mean curvature H. For a positive pressure p, the converse holds.

[0054] One embodiment achieves the flexible membrane pre-stretching by positioning the centerlines y above the membrane boundary 218 by a specific height h0. This pre-stretching process involves inflating the flexible membrane above the specific height ho and gradually deflating it. As a first approximation, the pre-stretching derives from the extension achieved when the membrane contacts the centerlines y during deflation. For specified coefficients A and p, the assumption p ~ A, K±~ A, K2holds. Assuming a spherical inflated flexible membrane, setting K±= K2= VQ, where Q represents the sphere radius, leads to Q = Jp. The height ho follows from the sphere radius Q and the minimum membrane span d, according to:

[0055] A Weingarten surface satisfying Equation (6) serves as a suitable approximation for a region on the membrane surface Q matching the target panel T. Cutting the offset surface 420 (obtained from the Weingarten surface 404) with four planes 216I at a set distance from the panel 402 boundaries (the ends of the offset surface 420 in FIG. 4B) yields four intersection curves 218. These curves provide theinitialized pipe surfaces centerlines y, accounting for membrane thickness h and the radius r of the bullnose edges on the support panels 216I.

[0056] In one embodiment, the target panel 402 is described by a B-spline surface, which extends beyond the panel boundaries. However, the method discussed above works for any other representation (e.g., a dense triangle mesh) of the target panel. The optimization of this surface proceeds toward a Weingarten surface with a curvature relation prescribed by Equation (6). Intersecting the optimized surface with the four sides (panels 2161) of the support frame 214 supplies the initial guess (step (1) discussed above) for the pipe surface centerlines y. Utilizing the coefficients A and. from the Weingarten surface, the process determines a suitable height h0for the centroid of the curves y using Equation (7). Afterward, the curves y undergo a rotation around their centroid to minimize the support frame 214 height. The initial rigid pose (3?, t) of the target panel T results from aligning the panel 402 relative to the pipe surface centerlines y.

[0057] To illustrate this process, the inventors compared the surface extension results using a Non-Uniform Rational B-Splines (NURBS)-based approach, with NURBS representing a standard mathematical model in computational geometry and Computer-Aided Design (CAD) for complex, smooth, free-form curves and surfaces, with those from the Weingarten surface extension. The NURBS extension produces distortions and inaccuracies in regions of negative curvature, causing significant deviations from the intended geometry. In contrast, the Weingarten surface extension achieves greater accuracy and visual coherence, particularly for complex geometries. Only the Weingarten surface approach enables a reliable estimation of the prestressdistribution and the initial height h0. This process incorporates the underlying surface curvature, which allows precise optimization that minimizes error and better reflects the material’s physical behavior.

[0058] The discussion now turns to the inverse design with physical simulation. The initial geometric initialization (step (1) as discussed for FIGs. 4A and 4B) provides an initial estimate of the geometry of y, the estimated height h0for the boundary-panel pair (y, ), and an initial rigid pose (3?, t) of the panel. The geometry-only approach (step (1)) omits the physical parameters of the flexible membrane 110, such as Young’s modulus E and Poisson’s ratio v, resulting in discrepancies between the fabricated and target panels. Consequently, further physical optimization (step (2) mentioned above) becomes necessary to reduce the mean and maximum errors on the panel, using a simulation-based optimization. This approach leverages an accurate geometry-aware priors while achieving physically feasible designs.

[0059] For the simulation-based optimization (step (2)), the problem is naturally formulated as a pipe surfaces centerline optimization (i.e., optimizing the center line of the pipe surface), minimizing the distance between the panel T and the surface Q of the flexible membrane by adjusting the pipe surfaces centerlines y parameters. The inventors introduced a novel approach (instead of minimizing the distance between the panel and the surface of the flexible membrane) of reparametrizing the problem from pipe surface centerlines y optimization to a membrane Q shape optimization. Several practical reasons support this choice.

[0060] Regarding the no-slip contact between the membrane and the support frame, previous embodiments modeled the pipe surface centerlines used in panelfabrication with an extremely high friction coefficient, extending beyond the typical frictional contact simulation range in modern computer graphics. This model, grounded in physical experiment observations, accurately reflects the no tangential sliding and no penetration characteristics of the mold in the fabrication process.

[0061] Panel alignment complexity arises because, although the membrane Q can be considered a function of y, only the alignment of panel T to Q is of interest. Employing a barrier formulation introduces complexity due to the choice of barrier thickness (commonly around 1 mm, or, if considering the size of the target shape, about 1 / 20 to 1 / 100 of the target panel diagonal), as the distance energy varies with millimeter-level offsets.

[0062] Penalty / barrier force modeling, while common for differentiable contact, proved numerically unstable in this setup when accumulating gradients from the panelmembrane proximity energy to boundary curves, even with an implicit simulator, unless using impractically small time steps.

[0063] Therefore, instead of optimizing y, this embodiment minimizes the distance (Equation (1 )) between the membrane Q and the panel T by adjusting points x e Q and the panel rigid pose (3?, t). This approach simultaneously (a) optimizes the rigid pose of T to align with membrane Q, and (b) modifies the shape of Q to match T. The membrane Q, governed by elastic energy, transforms the shape optimization problem from a traditional rigid alignment (registration) problem. Equation (1) frames the problem as a rigid panel T and a deformable surface Q, splitting the distance minimization into two sub-problems in an as-rigid-as-possible (ARAP) style: (i) panel rigid alignment, and (ii) membrane surface shape optimization.

[0064] For sub-problem (i), the deformable membrane surface Q remains in stasis. The rigid alignment problem optimizes the panel rigid pose (3?, t), that is, the rotation and / or translation of the panel, to minimize the proximity objective 2) (Equation (1)):min | ||5?y + t — projn(y)||2cL4X6SO(3),t61R3JyeTIIS.t. 0(2?)y= 0, txz= 0. (8)

[0065] In this formulation, dA represents an infinitesimal area, and 9 is a mapping from SO(3) to K3, associating a rotation matrix 5? with a vector of Eulerian angles in HR3. This embodiment uses a y-up coordinate system, so the constraints prevent changes in y-axis rotation and x, z-axis translation, limiting the rigid pose of T to three degrees of freedom and ensuring the panel remains within a bounding box B defined by the pipe surfaces centerlines y.

[0066] For sub-problem (ii), the rigid pose of T remains fixed, and the optimization focuses on the shape of Q to minimize the distance in Equation (1), subject to elastic energy constraints. As with Equation (8), the left inverse of proj_Q(-), denoted proj^Q, maps a point y e projq(y) to a point x e T. The shape optimization problem becomes:

[0067] Here, f represents the system force applied to Q, so f = 0 denotes a nodal force equilibrium state (Equation (3)).

[0068] One skilled in the art would understand that the above discussed algorithm is only one way to obtain the border curves for the membrane. Another optionwould be a data-driven approach where plural forward simulations are used to train a model that predicts the border curves. Further, based on the embodiments discussed in this document, one skilled in the art would understand that it is possible to control an aspect (e.g., shape) of a system (e.g., the deflated membrane 110) by controlling its boundary conditions (e.g., supports). This approach needs to solve an inverse problem for controlling the boundary conditions.

[0069] Method 500, which is illustrated in FIG. 5, outlines the fabrication of a doubly-curved panel based on the above embodiments. The method includes a step 502 of receiving a target curvature profile of the doubly-curved panel; a step 504 of determining boundary support curves for a flexible membrane to mimic the target curvature profile, where these curves define a surface larger than that of the target curvature profile; a step 506 of placing a support frame inside a housing, with a top perimeter of the support frame defining the boundary support curves; a step 508 of attaching the flexible membrane to the housing; a step 510 of applying a deflation pressure within the housing so that the flexible membrane deflates to follow the boundary support curves, thereby forming a mold mimicking the doubly-curved panel; and a step 512 of forming (e.g., casting) a material over the flexible membrane to form the doubly-curved panel.

[0070] In one embodiment, the target curvature profile specifies a non-standard, three-dimensional, free-form geometry or shape of the doubly-curved panel, which may derive from a design surface such as a B-spline surface. Determining the boundary support curves may involve using a Weingarten surface to model a target surface of the panel or the B-spline surface encompassing the target curvature profile, extracting pipesurface centerlines from an intersection of the Weingarten surface with a plane that is central to the support frame, to define initial boundary support curves, and establishing an initial rigid pose for the doubly-curved panel from the extracted centerlines.Selecting the pre-stretching height for the pipe surface centerlines may rely on a geometric simulation of the flexible membrane.

[0071] The method may further minimize a proximity objective between the initial rigid pose of the doubly-curved panel and the flexible membrane’s surface to obtain a final rigid pose and minimize the distance between this final pose and the deformed membrane shape to determine the boundary support curves and pre-stretching height. The deformed flexible membrane shape depends on the flexible membrane’s material properties, including elasticity, thickness, and boundary force response.

[0072] The method may include raising the support frame by the pre-stretching height to achieve the desired pre-stretching of the flexible membrane. The support frame includes three or more planar members, each ending in a bullnose edge that defines the top perimeter and corresponds to a boundary support curve, with each boundary curve differing from the others.

[0073] The method 500 was implemented by the inventors for experiments and testing utilizing the reconfigurable mold system 200. The system 200 employed a 3 mm thick silicone flexible membrane 110 with Shore A hardness of 40 and a density of 1190 kg / m3. The flexible membrane 110 was clamped in a two-part machined aluminum frame 220 with inner dimensions of 660 mm x 880 mm x 3 mm (width x depth x thickness). The frame 220 was positioned on an almost airtight aluminum-profile housing (box) 202, with dowel pins ensuring correct placement.CNC-machined medium-density fiberboard (MDF) support planes 2611 forming the support frame 214 were installed inside the housing 202. Lowering the air pressure inside the housing 202 caused the membrane 110 to deflate and contact the MDF support frame 214. The slightly rough, rounded edge of the MDF boards 216I prevented the flexible membrane 110 from slipping during deflation. Air pressure reduction below ambient was achieved using the first pump 204, which in this embodiment included two pneumatic Venturi vacuum generators: one generator provided constant base airflow, while a fast-acting magnetic valve on the high-pressure side controlled the second generator. In a different embodiment, the first pump 204 may be implemented as a vacuum pump powered by an electric motor, a variable frequency drive to control the motor speed, one or more sensors for capturing the membrane’s shape and its change during deflation, and a controller (e.g., a RID controller 208) to control the pump. A closed-loop controller 208 regulated the flexible membrane 110 deflation by monitoring a distance with a laser triangulation sensor 224. Although the internal pressure difference p within the housing 202 was monitored, the system 200 did not use this pressure as a control variable. Instead, the flexible membrane deflation was governed by the laser sensor 224’s distance measurement rather than pressure, thereby mitigating error stemming from material parameter measurements. For example, a 10% variation in Young’s modulus during the optimization process produced nearly identical membrane shapes but significantly different pressures. Controlling the deflation via the flexible membrane’s shape measurement effectively eliminated this error source.Alternatively, another embodiment could use the measured pressure to regulate the housing pressure.

[0074] The support panels 2161 included, in this embodiment, CNC-machined MDF boards. Other materials are also possible. The manufacturing process for the support panels 2161 shapes the top edge of each support panel. Experimental panels were fabricated from both manually applied and robotically 3D printed alabaster gypsum. For architectural-scale applications, alternative materials such as concrete or fiber-reinforced laminate offer viable options.

[0075] The system 200 illustrated in FIG. 2 operates with any incompressible membrane material characterized by a Young's modulus (E) and Poisson's ratio (v) suitable for thin shell modeling. The physical properties of the selected flexible membrane material directly influence the deflation process and the final panel geometry. The inventors collected flexible membrane specifications, including a thickness of 3 mm and a density of 1190 kg / m3, from the manufacturer's datasheet. Calibration of the silicone membrane’s mechanical properties using a tensile testing machine yielded a Young's modulus E = 1.2MPa and a Poisson's ratio v = 0.5.

[0076] The geometric initialization stage (step (1) discussed above) approximated the panel shape as a Weingarten surface. This method optimized a given B-spline surface (the initial panel geometry) toward a Weingarten surface by enforcing a specified relation between a mean curvature (H) and a Gaussian curvature K) at discrete sample points. The Levenberg-Marquardt algorithm served as the optimization method, expressing constraints and target functions as a system of quadratic polynomial equations solved iteratively. At each sample point, thealgorithm introduced variables A,p,p, along with auxiliary variablesand K2= K2|K2|, and corresponding auxiliary constraints:where the subscript * indicates values from a previous iteration, and the relationshipKI,2 = H ± VW2- K was applied. The target function in Equation (6) was enforced as A + ic2) + 2 pH + p = 0 at each sample point. The method oriented the membrane surface normal toward the open side, ensuring the vacuum produced a constant negative pressure p. To guarantee positive coefficients (A, p) and a negative pressure p, the algorithm added constraints c - |c*| = 0, with c = A,p, -p. The optimization used fourth-degree B-spline surfaces with eight control points per parameter direction. Computation times for test cases ranged from 4.47 s to 26.03 s.

[0077] The physical simulation phase (step (2) discussed above) involved discretizing the membrane Q into a triangular mesh {V, E, F}, where V e ]R|y|x3denotes vertices, E e lRl£lx2denotes edges, and F e lRlFlx3denotes faces. The membrane’s shape is considered to be the relaxed shape of membrane. This rest shape can be considered to be planar. In one embodiment, the shape of the membrane is obtained by capturing (3D scanning) the actual rest shape, preferably every time before the fabrication of a pane. The boundary vertices of Q were designated as dQ. The reconfigurable mold simulator utilized the thin shell simulation library LibShell [Chen et al. 2018]. All experiments employed a St. Venant-Kirchhoff (StVK) material model and the mid-edge average formulation for bending energy discretization. The StVK model’s simplicity and sparser Hessian structure facilitatedefficient sensitivity analysis. However, one skilled in the art would understand that other Isotropic Hyperelastic Models (such as Neo-Hookean, Mooney-Rivli, Yeoh, Ogden or the Arruda-Boyce model) or Anisotropic Hyperelastic Models (such as Gasser-Ogden-Holzapfel model) or even Viscoelastic Models (to account for creepage of the membrane) can be used depending on the specific material properties of the membrane.

[0078] The simulation defined a discrete pressure force fp(see Equation (3)) as:where p is the pressure constant, xtis the i-th vertex in Q, N(xi) is the set of faces sharing xt, Aeis the area of face e, and neis the unit normal of face e.

[0079] To initialize the pressure constant p in Equation (10), the process minimized the L2distance between the panel T and the membrane Q (Equation (1)) to find the optimal pressure constant p*p* = argmin‘D(p). (11) p

[0080] A forward quasi-static (also dynamic is possible) simulation was then executed, and sensitivity analysis related Ap to Ax. The deflation process was modeled as a quasi-static simulation, assuming slow membrane deflation to maintain equilibrium at every step and omitting the inertial term from the equation of motion. The simulation iteratively solved the following equation until convergence:where Vf is the ]R3|y|x3|y|force Jacobian, a e HR is the descent rate, and dy(-) is the distance to pipe surface y, constrained to exceed the constant radius R = r + h / 2 (see FIG. 4A). The simulation enforced this constraint throughout the process. At each equilibrium state, xkwas subject to a pressure force with pk. The algorithm gradually increased p1:feuntil reaching the optimal value p*, minimizing the distance T) (Equation (1)). The Newton-Raphson method with backtracking line search solved the equation, and the Conjugate Gradient (CG) method addressed the linear system. The simulation terminated when the squared norm of the displacement ||<5x|| 2 fell below a predetermined threshold E.

[0081] The process incorporated collision detection and response by investigating various penalty force-based frictionless contact models. The simulation remained viable only with an extremely high penalty coefficient and small descent rate a, given the physical parameters (thickness h and pressure constant p).Attempts to stabilize the simulation with an implicit time integrator did not resolve the instability. Furthermore, the frictionless contact contradicted the need for the mold to support the panel’s weight, as any load would cause membrane sliding along y. Addressing both the physical design and the numerical challenges, the method implemented a geometry-level step-and-project: after advancing from one equilibrium state xk(with pressure pk) to the next xk+1(with pressure pk+1), the algorithm detected the distance dYbetween the support curve segments y and membrane vertices x, ensuring that the vertices moved perpendicular to the tangent plane and did not slide along it.

[0082] The method enforced Dirichlet boundary conditions on the vertices in contact with y, ensuring non-penetration and non-sliding contact constraints as specified in Equation (12). The reconfigurable mold system 100 employed a high friction coefficient on the pipe surface, effectively immobilizing the flexible membrane 110 upon contact. This approach ensured zero penetration along the contact normal and zero sliding along the contact tangent. This formulation, combined with curve fitting, provided high accuracy for the intended application.

[0083] The local-global optimization process, as described by Sorkine

[2007] , proceeded in two steps. The local step optimized the rigid pose of the panel T to minimize the proximity objective 2) (Equation (1)). The global step adjusted the membrane shape Q to further minimize 2), subject to the total force f equilibrium condition.

[0084] The procedure discretized the target panel T with a triangular mesh {V, 8, T}. During the local step, the algorithm assumed static equilibrium for the membrane Q. The rigid pose optimization of T used the Iterative Closest Point (ICP) method with a point-to-plane objective (Equation (8)):where xy. is the closest point on Q and nyiis the unit normal on Q at xy.. The method enforced rotation / translation constraints with a projected gradient descent, projecting the updated rotation 2? to the nearest admissible rotation 2?' in SO(3) and setting the translation t to zero along the x and zaxes.

[0085] For the global step, the algorithm posited that infinitesimal changes to pipe surface centerlines y preserved the set of contact vertices, affecting only their positions. After each forward simulation reaching a force equilibrium (Equation (3)), the method denoted the set of contact design vertices as xdc x e fl, and the set of free vertices as xf= x \ xd. A sensitivity analysis modeled the optimal change in the....,, dT>pipe centerline design Ay « Axd= —:

[0086] Upon completing a forward simulation, the method updated the design vertices xdwith Axd(Equation (14)) without rerunning the simulation. The gradient dD— represented the proximity objective’s gradient with respect to the free vertices xf.d x d x To compute — „ the algorithm constructed the sensitivity matrix 5 = from theforce Jacobian in Equation (12).

[0087] The incremental update equation in Sensitivity Contour related changes in the design vertices to free vertices via the static equilibrium equation (Equation (3)):where 5 is the ]R|Xd|xlx / l sensitivity matrix mapping Axzto Axd. The LDLT direct solver addressed this linear system.

[0088] The inventors observed that the gradient Axdexhibited strong localization at each optimization step, resulting in abrupt, spiky changes. These changes could not be interpreted as smooth modifications to the pipe centerlines Ay.To address this, a Laplacian smoothing filter was applied to Axdbefore updating the design vertices:where i is the descent rate, M is the mass matrix, L is the Laplacian matrix, and Asis the Laplacian smoothing hyperparameter, set to the average edge length of the triangular mesh Q in all experiments.

[0089] After applying the Laplacian smoothing to update the design vertices (Equation (16)), the method relaxed the updated shell configuration with a quasistatic simulation step (Equation (5)), maintaining the pressure at the optimal value p* and holding the rigid pose of the target panel T constant.

[0090] The method iterated the local-global optimization until convergence. After extracting polylines from the optimized Weingarten surface, the algorithm derived the final pipe surface centerlines y from the membrane Q. The fabrication process, as previously described with reference to FIG. 2, then produced the panel T using these centerlines.

[0091] The computational fabrication process was evaluated using four baseline test panels and a facade composed of nine panels with varying curvature properties. Standard practice in computational architectural design categorizes the panels by their curvature: positive Gaussian curvature (K > 0), negative Gaussian curvature (K < 0), and developable panels with zero Gaussian curvature (K = 0).

[0092] The four baseline test panels (TP) selected (see FIG. 6) provided a representative range of curvature and accuracy for the reconfigurable mold system 200 and method 500. The test panels (TP1-TP4) shared similar dimensions butdisplayed different curvature characteristics: TP1 with positive Gaussian curvature (K = 2.72 to 4.89); TP2 with higher positive Gaussian curvature (K = 4.33 to 20.71); TP3 with a saddle shape and negative Gaussian curvature (K = -12.12 to -0.80); and TP4 exhibiting a sign change in the Gaussian curvature (K = -3.78 to 3.76).

[0093] Simulation and fabrication results for these panels were obtained (not shown). Panels TP2 (K » 0), TP3 (K < 0), and TP4 (K > 0 and K < 0) demonstrated curvature properties typically challenging to fabricate using traditional molds. These panels also exhibited higher Gaussian curvature values than most architectural structures. For instance, TP2 reached a positive K of 20.71, compared to the maximum K of 3.49 for the fabricated facade, while TP3 reached a negative K of -12.12, compared to the facade’s minimum K of -4.41.

[0094] To assess panel accuracy, the evaluation sampled five frames logarithmically during the optimization process. Additional optimization steps drove the mold mid-surface to converge toward the target panel geometry. Errors concentrated at boundaries and corners, with boundary errors exceeding mean errors (mean absolute error [MAE] and mean absolute percentage error [MAPE] in Table 1, FIG. 7) throughout optimization for all panels. Table 1 presents quantitative evaluation forTP1-TP4, measuring MAE and MAPE both overall and in x, y, and z directions, as well as maximum absolute and relative errors in each spatial direction. The upper part of the table reports optimization results; the lower part of the table reports deflated membrane scan results.

[0095] A large reduction in mean and maximum distance errors occurred for TP2 and TP4, as the initial Weingarten surface fit exhibited high errors (up to 6 mm)due to the initial pose’s deviation from the best-fit surface. The local registration step in optimization rapidly reduced these errors, ultimately achieving mean errors around 1 mm — an 83% reduction for both panels. TP1, designed for close geometric alignment with the deflated membrane, showed minimal error reduction, as the initial errors were already sufficiently low for fabrication. The optimization for TP1 demonstrated a rapid convergence (fewer than 30 steps) for panels near the mold mid-surface, validating the optimization process. The negatively curved TP3 demonstrated a 63% decrease in mean error, from 1.05 mm to 0.39 mm. In the final optimization state, all panels exhibited relative errors under 1% compared to the target panel, as reported in Table 1, FIG. 7.

[0096] The facade design capabilities of the reconfigurable mold system 200, including scalability and robustness, provide advanced solutions for modern architectural applications. Modern buildings increasingly utilize facades composed of panels fabricated with unique molds. The following discussion presents a facade specifically engineered to exceed the complexity of real-world examples, using system 200 and method 500.

[0097] The engineered facade incorporates three challenging geometric properties: (1) positive and negative Gaussian curvatures, (2) a wide range of Gaussian curvatures at both positive and negative extremes, and (3) nearly developable panels. An almost developable surface defines a panel with Gaussian curvature | / C| < e, where e is significantly less than the mean Gaussian curvature of the facade.

[0098] A portion of the facade consists of nine panels arranged in a 3x3 array. The panel numbering follows a systematic order from top left to bottom left as P1, P2, P3, and from top left to top right as P1, P4, P6, with P5 occupying the center position. Out of the nine panels, seven panels exhibit mixed curvatures with distinct Gaussian curvature distributions, while one panel remains almost developable.

[0099] The objective in facade paneling involves constructing a coherent surface that remains smooth and continuous, with each panel accommodating the geometry of adjacent panels. Previous research on surface paneling required simultaneous optimization of adjacent panels to prevent kinks. The experiment performed with the reconfigurable mold system 200 addressed the challenge of optimizing each panel independently, then assembling a facade from these discrete panels. The manufactured panels (based on system 200 and method 500) achieved precise alignment.

[0100] Gypsum served as the fabrication material for model-scale panels in one embodiment. For larger-scale production, panels can be manufactured from diverse materials, provided processing temperatures align with the requirements of the flexible membrane. Suitable materials and material combinations for multimaterial panels include thermoplastics such as PMMA (acrylic glass) or PC (polycarbonate), cast materials like concrete and ceramics, and fiber / resin systems, including laminates made from glass fiber mats and epoxy resin. Integration with additive manufacturing technologies, including 3D printing on the flexible mold, yields panels with accurate, smooth surfaces and defined boundaries. The 3D printing process also enables controlled panel thickness and the integration of additionalfeatures, such as mounting interfaces. The material can either be applied onto the flexible membrane in its deflated, curved, state or prior to deflation, in its planer state. This also allows the use of planar prefabricated blanks. The range of materials can be expanded by choosing a different material for the flexible membrane, coating the membrane, using composite materials, adding internal structures, and / or additional layers to the membrane or by active heating or cooling. Adding a layer that provides thermal insulation and actively cooling the membrane allows the processing of hot materials like glass in its viscous state.

[0101] In one embodiment, the support frame 214 may be replaced by a limited number of digitally controlled pistons connected by a flexible member. The pistons may be positioned along the edges, occupying positions approximately 28 to 44.

[0102] The method 500 enables the fabrication of a broad set of target panels 풯 by deflating membrane 풮 in contact with support frame centerline 훾. The method 500 approximates target panel shapes by utilizing a subset of the membrane's shape space, assuming similarity with variations in 훾. Industry standards regard the 3×3 model as highly curved, although facade panels typically exhibit modest curvature due to the scale. Panels with consistent mean curvature normals suit the reconfigurable mold system 200, as the uniform curvature ensures smooth deformation, reduces tension imbalances, and supports accurate reproduction of curved surfaces.

[0103] Controller 208 may store method 500 as illustrated in FIG. 2. FIG. 8 presents a possible structure for controller 208, depicted as computing device 800.Hardware, firmware, software, or combinations thereof implement the described steps and operations. Computing device 800 performs the activities detailed in the preceding embodiments and may include server 801. Server 801 incorporates a central processor unit (CPU) 802 connected to random access memory (RAM) 804 and read-only memory (ROM) 806. ROM 806 may include other storage media types, such as programmable ROM (PROM) and erasable PROM (EPROM).

[0104] Processor 802 communicates with internal and external components via input / output (I / O) circuitry 808 and bus 810, providing control signals and related functions. Processor 802 executes diverse functions as determined by software or firmware instructions. Server 801 houses one or more data storage devices, including hard drives 812, solid-state drives 814, and other hardware for reading and storing information, such as DVD drives. Software for the described steps may reside on memory stick 816, solid-state storage device 818, or other portable media, which may be accessed by solid-state drive 814, disk drive 812, or similar devices. Server 801 connects to display 820, compatible with LCD, plasma, cathode ray tube (CRT), and other presentation screens. User input interface 822 comprises multiple mechanisms, including mouse, keyboard, microphone, touchpad, touchscreen, and voice-recognition systems.

[0105] Server 801 may connect to external devices such as scanners or other data imaging systems. Server 801 may operate within a larger network configuration, including global area networks (GAN) such as the Internet 828, to facilitate connections with landline and mobile computing devices.

[0106] Controller 208 may be embodied as computing device 800 or, in certain embodiments, as a chip or chipset. The controller 208 may include one or more physical packages (chips) comprising materials, components, and wires on a structural assembly (baseboard). The structural assembly provides physical strength, compactness, and electrical isolation for the included circuitry. Controller 208 thus may function as a single chip or system on a chip, serving as the means for executing one or more operations that enable the described functionalities.

[0107] Processor 802 may assume various forms, including coprocessors, microprocessors, controllers, digital signal processors (DSPs), processing elements with or without accompanying DSPs, or alternative processing circuitry such as application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), microcontroller units (MCUs), hardware accelerators, or special-purpose computer chips. In certain embodiments, processor 802 includes multiple processing cores configured for independent operation, supporting multiprocessing within a single package. Alternatively, processor 802 may comprise several processors connected via bus 810, enabling independent instruction execution, pipelining, and multithreading.

[0108] Processor 802 executes instructions stored in memory device 804 or accessible storage. Alternatively or additionally, processor 802 executes hard-coded functions. Hardware or software configurations, or combinations thereof, enable processor 802 to perform operations according to the described embodiments. For example, when configured as an ASIC, FPGA, or similar device, processor 802 implements the described operations through dedicated hardware. When acting as asoftware executor, processor 802 utilizes instructions that specifically configured its operation for the designated algorithms and processes. When implemented in a device such as a pass-through display or mobile terminal, processor 802 operates under further configuration to support the present invention’s algorithms and functions. Processor 802 includes essential elements such as a clock, arithmetic logic unit (ALU), and logic gates to support its functions.

[0109] The terms “about” and “substantially,” as used in this application, signify a variation of up to 20% of the stated parameter. The use of terms such as first, second, and similar descriptors serves only to distinguish one element from another, without implying order or limitation. For instance, an object or step designated as first could also be referred to as second, and vice versa, without affecting the scope of the disclosure. Both objects or steps remain distinct, regardless of their designation.

[0110] The terminology in this description serves the purpose of describing specific embodiments and does not impose limitations. Singular forms “a,” “an,” and “the” include plural references unless context dictates otherwise. The term “and / or” encompasses all possible combinations of the associated listed items. The terms “includes,” “including,” “comprises,” and “comprising” indicate the presence of specified features, integers, steps, operations, elements, or components without precluding the presence or addition of others. The term “if” may be interpreted as “when,” “upon,” “in response to determining,” or “in response to detecting,” depending on context.

[0111] The disclosed embodiments present a reconfigurable mold system and corresponding method for manufacturing curved panels. This description does not limit the invention. The embodiments encompass alternatives, modifications, and equivalents within the scope of the appended claims. The detailed description provides comprehensive understanding through specific details, but various embodiments may be practiced without all such particulars. Each feature or element described may be used independently or in combination with other features or elements, as appropriate.

[0112] This written description provides examples to enable skilled individuals to practice the disclosed subject matter, including device and system fabrication and method implementation. The patentable scope is defined by the claims and may include additional examples recognizable by those skilled in the art. Such examples are within the intended scope of the claims.

[0113] The entire content of all referenced publications is incorporated by reference in this patent application.B. Kromoser and J. Kollegger. 2015. Pneumatic forming of hardened concrete -building shells in the 21 st century. Structural Concrete 16, 2 (2015), 161 -171.G. Quinn and C. Gengnagel. 2014. A review of elastic grid shells, their erection methods and the potential use of pneumatic formwork. WIT Transactions on the Built Environment 136. / / doi.org / 10.2495 / MAR140111.J. Panetta, F. Isvoranu, T. Chen, E. Siéfert, B. Roman, and M. Pauly. 2021. Computational inverse design of surface-based inflatables. ACM Trans. Graph. 40, 4, Article 40 (2021), 14 pages.T. Fischer. 2012. Geometry Rationalization for Non-Standard Architecture. Architecture Science 5 (01 2012), 25-47.M. Eigensatz, M. Kilian, A. Schiftner, N. Mitra, H. Pottmann, and M. Pauly. 2010. Paneling Architectural Freeform Surfaces. ACM Trans. Graphics 29, 4(2010), #45,1-10. Proc. SIGGRAPH.D. Pellis, M. Kilian, H. Pottmann, and M. Pauly. 2021. Computational Design of Weingarten Surfaces. ACM Trans. Graph. 40, 4, Article 114 (jul 2021), 11 pages. / / doi.org / 10.1145 / 3450626.3459939.D. Pellis, M. Kilian, H. Wang, C. Jiang, C. Müller, and H. Pottmann. 2020.Architectural freeform surfaces designed for cost-effective paneling mold re-use. In Advances in Architectural Geometry.C. Rogers and W. Schief. 2003. On the equilibrium of shell membranes under normal loading. Hidden integrability. Proceedings of the Royal Society of London.Series A: Mathematical, Physical and Engineering Sciences 459, 2038 (2003), 2449-2462.Y. J. E. Chen, S. H. Sheen, U. M. Ascher, and D. K. Pai. 2020. SIERE: A Hybrid Semilmplicit Exponential Integrator for Efficiently Simulating Stiff Deformable Objects. ACM Trans. Graph. 40, 1, Article 3 (Aug 2020), 12 pages.D. L. Michels, V. T. Luan, and M. Tokman. 2017. A Stiffly Accurate Integrator for Elastodynamic Problems. ACM Trans. Graph. 36, 4, Article 116 (July 2017), 14 pages. / / doi.org / 10.1145 / 3072959.3073706.D. L. Michels, G. A. Sobottka, and A. G. Weber. 2014. Exponential Integrators for Stiff Elastodynamic Problems. ACM Trans. Graph. 33, 1, Article 7 (feb 2014), 20 pages, / / doi.org / 10.1145 / 2508462.O. Sorkine and M. Alexa. 2007. As-rigid-as-possible surface modeling. In Symposium on Geometry processing, Vol. 4. Citeseer, 109-116.Rist, F., Wang, Z., Pellis, D., Palma, M., Liu, D., Grinspun, E. and Michels, D. L., 2024. A Flexible Mold for Facade Panel Fabrication. ACM Transactions on Graphics (TOG), 43(6), pp.1-16.

Claims

WHAT IS CLAIMED IS:

1. A method (500) for making a doubly-curved panel (402), the method comprising:receiving (502) a target curvature profile of the doubly-curved panel (402); determining (504) boundary support curves (218) for a flexible membrane (110) for mimicking the target curvature profile, wherein the boundary support curves (218) define a surface that is larger than a surface defined by the target curvature profile;placing (506) a support frame (214) inside a housing (202), wherein a top perimeter of the support frame (214) follows the boundary support curves (218); attaching (508) the flexible membrane (110) to the housing (202); applying (510) a deflation pressure within the housing (202) so that the flexible membrane (110) deflates according to the boundary support curves (218) to form a mold that mimicks the doubly-curved panel (402); andforming (512) a material over the flexible membrane (110) to form the doubly-curved panel (402).

2. The method of claim 1, wherein the target curvature profile represents a non-standard, three-dimensional, free-form geometry or shape of the doubly-curved panel.

3. The method of claim 1, wherein the target curvature profile is derived from a design surface.

4. The method of claim 3, wherein the design surface is a B-spline surface.

5. The method of claim 1, wherein determining the boundary support curves comprises:using a Weingart surface to model an extended surface that includes the surface of the target curvature profile;extracting, from the Weingart surface, pipe surface centerlines that define initial boundary support curves; anddefining an initial rigid pose of the doubly-curved panel using the extracted pipe surface centerlines.

6. The method of claim 5, wherein using the Weingart surface comprises: selecting a pre-stretching height for the pipe surface centerlines.

7. The method of claim 6, wherein selecting the pre-stretching height: comprises selecting the pre-stretching height based on a geometric simulation of the flexible membrane.

8. The method of Claim 6, further comprising:minimizing a proximity distance between a target surface corresponding to the initial rigid pose of the doubly-curved panel and a surface of the flexible membrane to obtain a final rigid pose of the doubly-curved panel; andminimizing a distance between the final rigid pose of the doubly-curved panel and a shape of the deformed flexible membrane to obtain the boundary support curves and the pre-stretching heigh, wherein the shape of the deformed flexible membrane is a function of material properties of the flexible membrane.

9. The method of claim 8, wherein the material properties include elasticity, thickness, and a response to boundary forces.

10. The method of Claim 8, further comprising:raising the support frame with the pre-stretching height to provide a desired pre-stretching of the flexible membrane.

11. The method of Claim 1, wherein the support frame includes 3 or more planar members.

12. The method of Claim 11, wherein each planar member ends with a bullnose edge, that defines a top perimeter of the support frame and corresponds to the boundary support curves.

13. The method of Claim 1, wherein each of the boundary support curves is different from the others.

14. A reconfigurable mold system (200) for making a doubly-curved panel (402), the system comprising:a housing (202);a support frame (214) disposed within the housing (202), wherein the support frame has a top perimeter that follows boundary support curves (218) that define a surface that is larger than a surface of a target curvature profile of the doubly-curved panel (402), wherein the target curvature profile represents a non-standard, three-dimensional, free-form geometry or shape of the doubly-curved panel;a flexible membrane (110) that is attached to the housing so that the flexible membrane deflates according to the boundary support curves; anda first vacuum pump (204) fluidly connected to the housing to draw air away from the housing to deflate the flexible membrane.

15. The system of Claim 14, further comprising:another membrane configured to be attached to the housing and form a chamber with the flexible membrane.

16. The system of Claim 15, further comprising:a second vacuum pump fluidly connected to a chamber formed by the flexible membrane and the another membrane to draw air away from the chamber to press a casting material onto the flexible membrane.

17. The system of claim 14, wherein the support frame includes 3 or more planar members.

18. The system of claim 17, wherein each planar member ends with a bullnose edge, that defines the top perimeter of the support frame and corresponds to the boundary support curves.

19. The system of claim 14, further comprising:plural extendable pins configured to reshape the flexible membrane.

20. The system of Claim 14, wherein each of boundary support curves is different from the others.

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