Filters in topology optimization based on Helmholtz‐type differential equations

By Boyan S. Lazarov, Ole Sigmund. 2011

1,110 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/1956b2cd740a28f23d1c8b7516d273254a2cf846

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In Civil and Structural Engineering

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Length scale and manufacturability in density-based topology optimization (2016, methodology): “The main advantages compared to classical filters are the lower computational cost [2], the re-utilization of solvers developed for the state problems [93], relatively easy parallelization for large scale problems [1], utilization of different filter parameters in different parts of the design domai”
Topology optimization using a continuous density field and adaptive mesh refinement (2018, background): “22 For a given density value ρ ∈ [0, 1], the Young modulus of the softened material becomes E(ρ) = Emin + ρ(E0 − Emin), (4) where p ⩾ 1 and Emin is a minimum value to prevent the assembled stiffness matrix from becoming singular.”
Minimum Compliance Topology Optimization of Shell-Infill Composites for Additive Manufacturing (2017, methodology): “173 We use the so-called PDE-filter based on a Helmholtz-type partial differential equation [25].”
On filter boundary conditions in topology optimization (2017, methodology): “For the PDE-filter, which in theory has infinite support, the choice does influence the filtered field, however, numerical experiments have shown that for all practical purposes it is sufficient to use a padding width equal to the filter radius, and that it makes no difference whether Dirichlet or h”
Efficient topology optimization in MATLAB using 88 lines of code (2011, methodology): “The computational cost can be significantly reduced by employing an iterative solver (Lazarov and Sigmund, 2010).”
Homogenization-based stiffness optimization and projection of 2D coated structures with orthotropic infill (2019, positive): “Smoothing As basis for subsequent projection we use the Helmholtz-type PDE-based density filter (Lazarov and Sigmund, 2011), − ( R 2 √ 3 )2 ∇(2)φ̃+ φ̃ = φ.”
Topology optimization approaches (2013, background): “Consequently, consistent derivatives have to be derived using the chain rule (Bruns and Tortorelli 2001; Bourdin 2001; Lazarov and Sigmund 2011; Kawamoto et al. 2011; Andreassen et al. 2011).”
General topology optimization method with continuous and discrete orientation design using isoparametric projection (2015, background): “Also, 2 ambiguity that stems from the periodic nature of the angular representation is potentially problematic and prevents application of filtering schemes [26, 27], which may help to yield better solutions or control manufacturability.”
A survey of structural and multidisciplinary continuum topology optimization: post 2000 (2014, methodology): “Recently, Lazarov and Sigmund (2011) developed more efficient variations of the basic sensitivity and density filters based on the solution of Helmholtz-type differential equations.”
Physics-informed neural networks with hard constraints for inverse design (2021, methodology): “These design parameters are further smoothed by a Helmholtz-type ﬁlter [28] with a ﬁlter radius about double the minimum mesh length scale.”
Multimaterial topology optimization by volume constrained Allen–Cahn system and regularized projected steepest descent method (2014, background): “It is important to note that there are many works in which authors computed the gradient of the objective functional on an H(1) space, for instance see [64].”
Systematic design of 3D auxetic lattice materials with programmable Poisson’s ratio for finite strains (2018, methodology): “To smooth the intersection between two super-ellipsoids, the PDE filter [46] is employed with r = 0.”
Topology optimization of coated structures and material interface problems (2015, background): “The expression is omitted here but may be found in the respective papers (Lazarov and Sigmund (2011) for the PDE-filter and Wang et al.”
Topology Optimization of Turbulent Flows (2018, methodology): “The projection is applied on a PDE filtered density field [41].”
Undercut and overhang angle control in topology optimization: A density gradient based integral approach (2017, methodology): “The integral kernel size r in (5) effectively corresponds to 2 √ 3 times the filter size in the usual density filtering [33].”
Topology optimization in B-spline space (2013, methodology): “Thus, the B-spline filter provides an efficient filtering scheme similarly as partial differential equation (PDE) based filter [14, 15, 16], but without the need to solve the PDE.”