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Research Contents

Research Contents

Research Contents

I.Construction of novel theory of discrete surfaces
  1. Geometry of piecewise smooth surface. Extending the discrete differential geometry which usually adopts planar quadrilaterals as the shape elements, we define a class of piecewise smooth surfaces with the properties of both discrete and smooth surfaces. We introduce the basic notions such as the curvature, and construct methods for analysis, such as the discrete variational principle. In particular, taking the high needs in design from the viewpoint of productivity, we take the developable surfaces as the shape elements, which is characterized as the surfaces with zero Gaussian surface. We then develop the theory motivated by practical design.
  2. Theory of “aesthetic surfaces” that enables the efficient design of the artistic shape. We take the “aesthetic surfaces” as the shape elements which have been studied in the industrial design. We adopt the framework of the similarity geometry which is one of the Klein geometries. Based on the fact that the log-aesthetic curves (LAC), which is the basis of the aesthetic surfaces, are the similarity geometric analogue of the Euler’s elastic curves in the Euclidean geometry, we introduce the aesthetic surfaces in the framework of the surface theory in the similarity geometry.
  3. Theory of class of surfaces with geometrically good properties or mechanically superior characteristics.We clarify the mechanical properties of the minimal surfaces of other Klein geometries such as the Wilmore surface and utilize them in the design. Also, we characterize the surfaces with superior mechanical characteristics, for example, the surfaces spanned by the geodesics, by the variational principle, and clarify the geometric properties. Then we construct the basic infrastructure to utilize them in the broader range of design.
II.Construction of the method of design based on the mathematical framework of I. and development of the design platform of mutual circular type.
  1. Novel parametric representation of surfaces and the method of generating piecewise smooth surfaces satisfying the geometric constraints. We establish the method of surfaces based on the discrete variational principle for the aesthetic fairness metric and the mechanical energy.
  2. Method of design for the surface architecture with aesthetic shape.We develop the method to design of a class of surfaces lying between the developable surfaces and the minimal surfaces, and the polygon shape such as the rigid-foldable roof close to the developable surface. In particular, we construct a technique to solve the optimization problem of the surface shape by the formulation of a multi-purpose optimization problem by using the notion of the distance between discrete surfaces obtained from the notion of the curvature for the piecewise smooth surfaces.
  3. Analysis, identification and optimization methods based on the geometric structure specified by the characteristic quantities for the piecewise smooth surfaces.We establish a novel method of predicting the deformation response for the structure with the loads as functions of the design variables. We also construct a method for the data-driven problem to predict the deformation from the experiment data of the materials directly by using the notion of the distance between the discrete surfaces.
  4. Surface structure with good geometric properties and with superior productivity and mechanical rationality.
  5. Application of discretization of geometric functional to the finite element method.We formulate a novel finite element method using the discrete differential forms, and realize the simple formulation independent of the coordinate systems and the method of structure analysis.
  6. Klein geometry analogue of the log-aesthetic curves and their implementation.
  7. Technique to generate piecewise smooth surfaces based on the surface normal map image.We develop a technique to solve the inverse problem to recover the surface from the normal map (Gauss map) that is indispensable to generation of surfaces by using the piecewise developable surfaces.
  8. Construction of interactive design method for the developable shapes with the folding along the space curves.We develop a discrete representation of the geometric shapes with folds arising from the surface folding and its interactive design method. We develop a technique combining two approaches: an approach to specify the angle of folding and fold (regular objective type) , and another approach specifying the Gauss map (reverse objective type).
III.Pioneering the novel design and mathematical technology by linking interdisciplinary areas.
  1. Application of the curve folding in the origami engineering to temporary structures.Incorporating the curve folding, the cutting-edge technology in the origami engineering, to the architecture design, and utilize it for the design of temporary structures with superior portability and rigidity. It will contribute to the improvement of quality of life in disaster.
  2. Proposal of developable aesthetic surface and membrane aesthetic surface and their application to architecture and shipbuilding.
  3. Continuous deformation of developable surfaces and Kleing geometric analysis. We analyze the developable surfaces with the Gauss-Codazzi equation and construct a theoretical tool to investigate the transition process of the folds by examining the solutions around the singularities (folds). Based on this, we describe the deformation of the developable surfaces by using the technique of Klein geometry. We then generate new classes of surfaces which lie between the developable surfaces and minimal surfaces by investigating the surface area minimization problem.

Research System

We organize the following research groups and carry out the researches based on the collaborations among the groups.

Research Group・Members

Kajiwara Group

Leader:
Kenji Kajiwara (Kyushu University) (representative)
Main research items:
Novel discrete surface theory.I. (a), (b), (c)

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Ohsaki Group

Leader:
Makoto Ohsaki (Kyoto University)
Main research items:
Design and optimization of piecewise smooth surfaces by using the variational principle of discrete differential geometry. II.(a), (b), (c)

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Honma Group

Leader:
Toshio Honma (Kagoshima University)
Main research items:
Architecture structure with superior productivity and mechanical rationality.
II. (d), (e), III. (a)

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Miura Group

Leader:
Kenjiro T. Miura (Shizuoka University)
Main research items:
Formulation of free curves and surfaces by Klein geometry and application to practical use.II. (f), (g), III. (b)

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Maekawa Group

Leader:
Takashi Maekawa (Waseda University)
Main research items:
Generation of piecewise developable surfaces based on the surface normal map images. II. (h)

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Mitani Group

Leader:
Jun Mitani (University of Tsukuba)
Main research:
Design of developable shapes with folding along the space curves.
II. (i), III. (c)

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Research Process

  1. The leaders meet at the strategic meeting once in three months each fiscal year to share the research information and adjust the research advancement. The members can also join if they wish.
  2. Among four meetings in each year, two of them are the general meetings where all the members participate.
  3. One of the above meeting will be the international workshop where the overseas members join.
  4. In addition, in the first year we will organize the kickoff meeting, and the international conference in the final year.
  5. In the 2020-21 or 2021-22 fiscal year, we establish “Applied differential geometry activity group” (preliminary) in Japan Society of Industrial and Applied Mathematics (JSIAM) to secure the opportunity for the activities of members including the young researchers such as the graduate students and postdocs.

The pre-history of this project

Traditionally, in Japan, pure mathematics oriented researches has been dominating in the mathematics community, and mathematician had almost no chance for a long time to contact various scientific areas and industry except theoretical physics. Institute of Mathematics for Industry, Kyushu University (IMI) was established in 2011 in order to change this situation and to promote the collaborated researches with industry and scientific areas. Kajiwara, the representative of this project, changed the fields from mathematical physics, in particular, discrete integrable systems to discrete differential geometry around 2009, and has been constructing the theoretical infrastructure through the studies of integrable deformation theory of discrete curves and the discrete holomorphic functions after appointed to IMI in 2011. In parallel, he has organized/participated in the following joint research programs of IMI and colloquia to search for the contact with the applied areas.

  1. Short-term joint research ”Differential and discrete differential geometry for design” (2016.9.28-30)
  2. Short-term joint research “Proposal of three dimensional geometric modeling evaluation method and software development” (2017.9.4-8)
  3. Short-term joint research ”New development of discrete differential geometry: from industrial design to architecture design”(2018.9.10-13)
  4. Short-term joint research “Surface geometry in shipbuilding” (2018.12.10-28)
  5. IMI Colloquium ”Geometric problems in architecture design” Makoto Ohsaki (2019.3.13)
  6. Short-term joint research ”Application of discrete differential geometry to design: from theory to practical use” (2019.9.9-11)
  7. AIMaP workshop ”Linking discrete differential geometry and finite element method: application to architecture and CG” (2020.3.6-8)

We started the collaboration with Miura by (1) regarding the geometry of aesthetic shapes. By (3), we started the joint work on the general discrete differential geometry, including Miura, Honma, and Yokosuka. As to architecture design, the Kajiwara group invited Ohsaki for (5) and started joint researches. Kajiwara and Kaji participated in (2) and (4) to learn the problems in the shipbuilding. (6) has become the 0-th strategic meeting essentially, and most of the leaders participated. We plan to arrange (7) as the kick-off meeting. Konrad Polthier, who is a world-level leading researcher in discrete differential geometry and is a member of the Kajiwara group, has been collaborating with Kajiwara and Koiso. In fact, Jikumaru and Park, the graduate students of the Kajiwara group, have experiences to visit Polthier for several months for joint researches. Wolfgang K. Schief, who is one of the pioneers of integrable discrete differential geometry, has been doing joint research on the aesthetic shape since 2016. Independently, Mitani, Maekawa, and Miura have close research interactions. This research project is carried out based on those activities.

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