American Institute of Mathematical Sciences

Advanced
October  2015, 8(5): 933-951. doi: 10.3934/dcdss.2015.8.933

Truss structure design using a length-oriented surface remeshing technique

Karol Mikula 1, , Mariana Remešíková 1, and  Peter Novysedlák 2,

1. 

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 81368 Bratislava, Slovak Republic, Slovak Republic
2. 

Hutnícka 10/51, 05201 Spišská Nová Ves, Slovak Republic

Received  December 2013 Revised  June 2014 Published  July 2015

We present a method that can be used for designing truss structures representing either minimal surface shapes or general free-form shapes. The structures are designed so that they meet some specific criteria concerning their aesthetic properties and especially the lengths of the truss elements. We explain a technique for tangential redistribution of points on evolving surfaces that allows to obtain equally sized truss elements in selected subsets of the structure. This technique is applied to surfaces evolving by their mean curvature yielding constructions that approximate minimal surface shapes. Afterwards, we show how to remesh static free-form surfaces.
Keywords: surface remeshing, tangential redistribution., Truss structure design, minimal surfaces, surface evolution.
Mathematics Subject Classification: Primary: 65M08, 65M50; Secondary: 53A0.
Citation: Karol Mikula, Mariana Remešíková, Peter Novysedlák. Truss structure design using a length-oriented surface remeshing technique. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 933-951. doi: 10.3934/dcdss.2015.8.933
References:
[1]

M. R. Barnes, Form finding and analysis of tension structures by dynamic relaxation, International Journal of Space Structures, 14 (1999), 89-104. doi: 10.1260/0266351991494722.

[2]

K. U. Bletzinger, M. Firl, J. Linhard and R. Wüchner, Optimal shapes of mechanically motivated surfaces, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 324-333. doi: 10.1016/j.cma.2008.09.009.

[3]

M. Húska, M. Medĭa, K. Mikula, P. Novysedlák and M. Remeší ková, A new form-finding method based on mean curvature flow of surfaces, Proceedings of ALGORITMY 2012, 19th Conference on Scientific Computing, Podbanské, Slovakia, September 9-14, 2012, Publishing House of STU, (2012), 120-131.

[4]

M. Meyer, M. Desbrun, P. Schroeder and A. H. Barr, Discrete differential geometry operators for triangulated 2-manifolds, Visualization and Mathematics III (Hans-Christian Hege and Konrad Polthier, eds.), 3 (2003), 35-57.

[5]

B. Maurin and R. Motro, The surface stress density method as a form-finding tool for tensile membranes, Engineering Structures, 20 (1998), 712-719. doi: 10.1016/S0141-0296(97)00108-9.

[6]

K. Mikula, M. Remeší ková, P. Sarkoci, D. Ševčovič, Surface evolution with tangential redistribution of points,, to appear in SIAM Journal of Scientific Computing., (). 

[7]

K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM Journal on Applied Mathematics, 61 (2001), 1473-1501. doi: 10.1137/S0036139999359288.

[8]

K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of planar curve with an external force, Mathematical Methods in Applied Sciences, 27 (2004), 1545-1565. doi: 10.1002/mma.514.

[9]

K. Mikula and J. Urbán, 3D curve evolution algorithm with tangential redistribution for a fully automatic finding of an ideal camera path in virtual colonoscopy, Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, 6667 (2012), 640-652. doi: 10.1007/978-3-642-24785-9_54.

[10]

A. Mottaghi Rad, H. Jamili and S. A. Behnejad, Length equalization of elements in single layer lattice spatial structures, Abstract Book of the IASS-APCS 2012 Conference, Seoul, Korea, (2012), 266-273.

[11]

J. C. C. Nitsche, A new uniqueness theorem for minimal surfaces, Arch. Rat. Mech. Anal., 52 (1973), 319-329. doi: 10.1007/BF00247466.

[12]

F. Pantano and H. Tamai, Geometric Multi-objective Optimization of Free-form Grid Shell Structures, Abstract Book of the IASS-APCS 2012 Conference, Seoul, Korea, (2012), 85-93.

[13]

R. M. O. Pauletti and P. M. Pimenta, The natural force density method for the shape finding of taut structures, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 4419-4428. doi: 10.1016/j.cma.2008.05.017.

[14]

U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates, Experim. Math., 2 (1993), 15-36. doi: 10.1080/10586458.1993.10504266.

[15]

H. J. Scheck, The force density method for form finding and computation of general networks, Computer Methods in Applied Mechanics and Engineering, 3 (1974), 115-134. doi: 10.1016/0045-7825(74)90045-0.

[16]

B. H. V. Topping and P. Ivanyi, Computer Aided Design of Cable Membrane Structures, Saxe-Coburg Publications on Computational Engineering, 2008.

show all references

References:
[1]

M. R. Barnes, Form finding and analysis of tension structures by dynamic relaxation, International Journal of Space Structures, 14 (1999), 89-104. doi: 10.1260/0266351991494722.

[2]

K. U. Bletzinger, M. Firl, J. Linhard and R. Wüchner, Optimal shapes of mechanically motivated surfaces, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 324-333. doi: 10.1016/j.cma.2008.09.009.

[3]

M. Húska, M. Medĭa, K. Mikula, P. Novysedlák and M. Remeší ková, A new form-finding method based on mean curvature flow of surfaces, Proceedings of ALGORITMY 2012, 19th Conference on Scientific Computing, Podbanské, Slovakia, September 9-14, 2012, Publishing House of STU, (2012), 120-131.

[4]

M. Meyer, M. Desbrun, P. Schroeder and A. H. Barr, Discrete differential geometry operators for triangulated 2-manifolds, Visualization and Mathematics III (Hans-Christian Hege and Konrad Polthier, eds.), 3 (2003), 35-57.

[5]

B. Maurin and R. Motro, The surface stress density method as a form-finding tool for tensile membranes, Engineering Structures, 20 (1998), 712-719. doi: 10.1016/S0141-0296(97)00108-9.

[6]

K. Mikula, M. Remeší ková, P. Sarkoci, D. Ševčovič, Surface evolution with tangential redistribution of points,, to appear in SIAM Journal of Scientific Computing., (). 

[7]

K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM Journal on Applied Mathematics, 61 (2001), 1473-1501. doi: 10.1137/S0036139999359288.

[8]

K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of planar curve with an external force, Mathematical Methods in Applied Sciences, 27 (2004), 1545-1565. doi: 10.1002/mma.514.

[9]

K. Mikula and J. Urbán, 3D curve evolution algorithm with tangential redistribution for a fully automatic finding of an ideal camera path in virtual colonoscopy, Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, 6667 (2012), 640-652. doi: 10.1007/978-3-642-24785-9_54.

[10]

A. Mottaghi Rad, H. Jamili and S. A. Behnejad, Length equalization of elements in single layer lattice spatial structures, Abstract Book of the IASS-APCS 2012 Conference, Seoul, Korea, (2012), 266-273.

[11]

J. C. C. Nitsche, A new uniqueness theorem for minimal surfaces, Arch. Rat. Mech. Anal., 52 (1973), 319-329. doi: 10.1007/BF00247466.

[12]

F. Pantano and H. Tamai, Geometric Multi-objective Optimization of Free-form Grid Shell Structures, Abstract Book of the IASS-APCS 2012 Conference, Seoul, Korea, (2012), 85-93.

[13]

R. M. O. Pauletti and P. M. Pimenta, The natural force density method for the shape finding of taut structures, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 4419-4428. doi: 10.1016/j.cma.2008.05.017.

[14]

U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates, Experim. Math., 2 (1993), 15-36. doi: 10.1080/10586458.1993.10504266.

[15]

H. J. Scheck, The force density method for form finding and computation of general networks, Computer Methods in Applied Mechanics and Engineering, 3 (1974), 115-134. doi: 10.1016/0045-7825(74)90045-0.

[16]

B. H. V. Topping and P. Ivanyi, Computer Aided Design of Cable Membrane Structures, Saxe-Coburg Publications on Computational Engineering, 2008.

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