Application of Layout and Topology Optimization Using Pattern Gradation for the Conceptual Design of Buildings Research Paper

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Current Research in Topology Optimization: An Article Review

Current research and practice in engineering and design relies increasingly on computation-driven optimization techniques that result in a more efficient and more effective use of resources, creating stronger and more lasting structures out of less materials. The computational problems that can and do arise in light of the complex tasks that current methodologies are called on to perform, however, have led to some significant limitations on the applicability of topology optimization results to real-world building projects. That is, the modeling completed by many topology optimization techniques performs well in theoretical applications but has proven difficult to control for the practical problems encountered in actual construction, leading to both underutilization in practical spheres and to intensive focus but not necessarily a great deal of headway in the research community. New techniques and methodologies continue to be developed to increase the effectiveness and the practicality of topology optimization techniques, however, and some have begun to show a great deal of promise when it comes to tackling certain specific design and engineering problems.

The following review examines a recent piece of published research in which the authors lay out their method for mapping different design variables in a consistent and cohesive manner that accounts for resource efficiency in addition to other design variables and constraints. Of specific interest to Stromberg et al. (2011) in "Application of layout and topology optimization using pattern gradation for the conceptual design of buildings" are varying material densities and the use of patterns to model and ultimately optimize design layouts -- a problem that has proven difficult to tackle in a means that yield practical and buildable results. Using a concept called pattern gradation and the multiresolution topology optimization technique or MTOP, the authors developed algorithms for topology optimization that maximized the stiffness of structures while also minimizing material usage, achieving practical results for a primary engineering problem that also contend with the practical realities of real-world building situations. Beginning with an extensive theoretical explanation of their technique and the parameters and equations involved, making a substantial case for this technique's likely success, the authors then describe the practical issues in high-rise building mechanics that their research applies to. Acknowledging that all of the assumptions made in their model will not necessarily translate to high-rise building, they nonetheless present a clear means of addressing these assumptions and for future research to further refine the model. The authors then provide the numerical results of their model's throughout and conclude by recapping the importance of pattern gradation in current practical applications as well as in ongoing research.


Stromberg et al. (2011) examine the "layout problem" primarily as it applies to high-rise buildings and the stiffness of these structures, and specifically examine the models of topology optimization and specifically of multiresolutional topology optimization as it applies to these structures of interest. Manufacturing expense and difficulty have proven too complex to account for in many models, with designs for the spatial arrangement of structural material given a variety of other more pressing variables that take precedence in most modeling techniques. Material patterns require certain densities based on structural needs, however the patterns also lead to an overuse of material in certain scenarios and repetitions. With pattern gradation used to inform and constrain the algorithms of topology optimization, the authors demonstrate that it is possible to use topology optimization methods to create more efficient buildings and structural patterns. Other elements that the authors note as of especial importance in refining current topological optimization techniques include dealing with longitudinal and torsional periods, incorporating room for utility pass-throughs (water pipes, electrical conduit, etc.), and the need to incorporate other materials (glass sheeting, concrete, etc.) that are not part of the structural element of the building. All of these issues, according to the authors, can be addressed through the use of pattern gradation in topology optimization in structural design.

Turning to the central problem of structural/building stiffness, the authors present a basic compliance equation set to define the density and stiffness of a structure, then adjust this equation such that density does not take on a zero (void) or one (material) value, but rather takes on a decimal value between zero and one (inclusive) to allow for the adjustment of density and material outlay. A basic outline of the MTOP methodology is also presented, describing the three mesh layers of displacement, element densities, and design variables and the manner in which they are used in tandem to deliver more practically applicable results than through standard topology optimization. Validation of the MTOP approach showing higher levels of structural compliance are also provided, giving a clear conceptual and numerical explanation of the methodology.

Nguyen et al. (2010) developed the MTOP technique specifically as a way to begin mapping material densities in layers with design variables and finite elements, creating three "meshes" of mapping and calculation that when used processed with optimization yield more refined information with a different and more comprehensive objective focus. The following figures taken from Nguyen et al. (2010) show the basic approach to the problem in a visual manner:

Layering the various elements/considerations of each building point in these individual yet superposed meshes results in a single multilayered mesh that yields more detailed optimizations and design capabilities, as demonstrated by further explanation and visual information presented by Nguyen at al (2010):

The multiple variables and constraints for each point are used to map out elements with greater accuracy and efficiency and in a relatively straightforward manner that can be carried out with known techniques.

Optimizing the completed element map yields structural design elements such as those pictured here, with the density changes and resource efficiency sought by Stromberg et al. (2011) already becoming apparent. The authors also utilized a continuous approximation of material distribution (CAMD) method in the MTOP element mapping, again following Nguyen et al. (2010), however the pattern gradation of density variables alters the outcome of the CAMD technique significantly.

After defining these foundational aspects of the theories and applications discussed throughout the paper, Stromberg et al. (2011) at last turn to the concept of pattern gradation that is central to their methodology. In light of the foundational theories discussed, Stromberg et al. (2011) identify pattern gradation as another constraint in layout optimization, similar in certain ways to the issues of materials needs and other design variables mentioned above. Frameworks for laying out design variables in layers of primacy are presented, with pattern gradation apparent in the outlay of the variables and in variable form as well, constructing the element map and thus the optimization model in such a way as to illustrate the pattern gradation first in a single direction and then in multiple directions across the computational model. Patterns are sized and scaled based on the domain sizes of the map and of the structure described, and formulas for individual points and design variables are presented in building the full model ultimately used in the topological optimization application. Throughout the mapping process and the later topology optimization, each secondary design variable's impact on the primary design variables is summed for each node of the element, and complex pattern gradation is achievable through the layering of the MTOP methodology, resulting in highly individualized calculations for each mapped element that nonetheless are optimized for both building material efficiency and ease of construction, attempting to maximize resource efficiency over the life of the building from construction through occupation.

The authors then apply the same theoretical models of pattern gradation in element mapping and topological optimization to some of the special circumstances defined earlier, such as holes in beams to allow for utilities to be run through structural elements, demonstrating the flexibility of the pattern gradation approach to solving problems of variable structural density and material usage. Variable influence can also be incorporated into the model without disrupting pattern gradation due to the modifications to the MTOP model and the element mapping calculations suggested by these authors. The model and methodology thoroughly described and calculated by the authors is demonstrated to be quite robust and broadly applicable to real-world situations, at least in the theoretical applications and operations that the authors themselves describe. Though these applications and the machinations of the methodology developed by Stromberg et al. (2011) are relatively complex, the underlying principles of the methodology are surprisingly simple given the degree of progress they represent: through the layering of design variables including a pattern gradient to control for varying densities and other properties as needed at various domain sizes, a high degree of specificity and control can be exerted over topological optimization outputs, resulting in more resource efficient and objectively focused design. The authors also present several constraint conditions that could be accommodated through further adjustments to the models and mapping calculations presented, and also demonstrating the ease with which weighting functions can be accomplished non-linearly in addition to through the linear functions the authors themselves present.

With this solid explanation and… [END OF PREVIEW]

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APA Format

Application of Layout and Topology Optimization Using Pattern Gradation for the Conceptual Design of Buildings.  (2012, December 10).  Retrieved February 16, 2019, from

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"Application of Layout and Topology Optimization Using Pattern Gradation for the Conceptual Design of Buildings."  10 December 2012.  Web.  16 February 2019. <>.

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"Application of Layout and Topology Optimization Using Pattern Gradation for the Conceptual Design of Buildings."  December 10, 2012.  Accessed February 16, 2019.