Composite lattice-skin structures combine the advantages of thin-shell and lattice structures. Owing to their curved geometry, shell structures exhibit a superior load carrying capacity and are lightweight, but respond extremely sensitively to any changes in the loading, geometry, etc. [8, 40]. In contrast, lattices are less sensitive, have more easily tunable mechanical properties, and are ideal for multi-functional designs that simultaneously draw on several properties, like stiffness, heat or inertia [19, 17]. Due to their favourable mechanical properties, lattice-skin structures are prevalent in engineering, e.g. lattice-core sandwich plates (Figure 5), and in nature, e.g. trabecular bone. Lattice and lattice-skin structures are experiencing a renaissance because of the rapid development of additive manufacturing, or 3D printing, technologies [18, 42]. Additive manufacturing makes it possible to create parts with nearly any shape and geometric complexity, hierarchically extending from the part scaled down to the resolution length of the printing process. To explore and exploit the resulting vast design space, computational approaches for rational design and optimisation of lattice-skin structures are indispensable.
Several powerful optimisation approaches have been proposed over the last 50 years for structures consisting only of lattices. Consequently, there is an extensive amount of literature that is impossible to review in detail. The ground structure method is one of the earliest techniques in the topology optimisation of pin-jointed truss structures [29, 3, 35, 54]
. It finds an optimal structure by optimising the cross-sectional areas of struts of a given highly-connected truss by usually solving a linear programming problem. The objective is to minimise the total material volume while satisfying the equilibrium and prescribed stress limits. Although in classical formulations the joint positions are fixed, some approaches consider the joint positions as additional optimisation degrees of freedom. Broadly, the ground structure method is not suitable for very large problems, even though there are methods to alleviate this limitation by incrementally adding members [20, 34, 22]. Beyond that, the ground structure method can lead to non-manufacturable thin struts and, crucially, it is unsuitable for optimisation of lattices connected to other structures. Classical homogenisation-based approaches for topology optimisation offer an alternative approach for optimisation of lattices consisting of repetitive unit cells [4, 49, 36]. In such methods, the lattice is first homogenised with the resulting effective material parameters depending on unit cell parameters, like the material content or cell orientation. These parameters are later determined by optimising the continuum structure composed of the homogenised material. Projecting back the obtained parameters to the unit cells gives an optimised lattice structure. In general, adjacent unit cells will not match across common cell boundaries, which has to be ensured by other means [49, 37, 21]. A more fundamental limitation of the homogenisation-based approaches is that the underlying assumption of scale separation breaks down, for instance, near boundaries and when the unit cell size is not sufficiently small compared to the overall size of the structure. The application of the two mentioned principal lattice optimisation techniques in additive manufacturing context has been explored, amongst others, in [36, 55, 41].
The shape optimisation of shell structures is an equally widely investigated topic. One of the key challenges in any kind of shape optimisation is to find a set of geometric design parameters that can parameterise a sufficiently rich set of geometries. To this end, classical geometry parameterisation techniques make use of splines [7, 23], filtered/smoothed finite element representations [30, 6], or free-form deformations [26, 43]. Lately, shape optimisation has greatly benefited from the ongoing academic and industrial interests in isogeometric design and analysis. In isogeometric analysis, the same spline representation is used for finite element analysis and geometric modelling . Given the unique refinability property of splines, a different resolution of the same spline representation can be used for parameterising geometry in optimisation [12, 28, 2, 24]. An additional benefit of smooth spline basis functions in shell analysis is that they are suitable for discretising the Kirchhoff-Love equations [11, 10, 33]. The resulting discretised equations depend only on the displacement of the shell mid-surface, simplifying the computation of sensitivities, or shape gradients.
The concurrent shape and topology optimisation of lattice-skin structures has been scarcely explored. There is somewhat related work on porous structures with a stiff outer coating [13, 14, 51, 50] and on lattice infill optimisation without taking into account the coupling between the lattice and the shell . To close this gap, we propose a concurrent shape and topology optimisation approach by combining topology optimisation of the lattice with shape optimisation of the entire structure. Building on our earlier work on isogeometric design and analysis of lattice-skin structures 
, the lattice is modelled as a pin-jointed truss and the skin as a Kirchhoff-Love shell. The lattice consists of a large number of cells, which in turn consist of a small number of struts connected by pins that do not transfer moments. This approximation is sufficient for the technologically important stretch-dominated lattice structures, see e.g.. The shell and the lattice are coupled using Lagrange multipliers. The topology of the lattice is optimised by taking the cross-sectional areas of the struts as design variables. To obtain an optimised black and white design, we penalise intermediate cross-sectional strut areas by choosing as design variables scaled cross-sectional areas as in the SIMP method . Furthermore, we introduce a cell-wise defined sensitivity filter to avoid scattered struts similar to checkerboards in continuum structures. The sensitivity filter, in combination with a postprocessing step for cell reconstruction, yields a lattice with no mechanisms . In shape optimisation, we consider the entire lattice-skin structure and parameterise its geometry using the free-form deformation technique, ensuring that the conformality between the lattice and the shell is always maintained. In computing the sensitivities for shape and topology optimisation, we take the mechanical coupling between the lattice and the shell fully into account. The topology and shape optimisation are performed sequentially. In our applications we first optimise the lattice topology and then the shape of both the shell and the lattice. Although we considered only compliance as a cost function, the proposed approach is straightforward to extend to other cost functions.
The outline of the paper is as follows. In Section 2, we briefly review the governing equations of lattice-skin structures and sketch their isogeometric finite element discretisation. Subsequently, in Section 3, we introduce the proposed lattice topology optimisation technique taking the coupling between the lattice and the skin into account. Building on that, in Section 4, after briefly introducing the free-form deformation technique for shape parameterisation, we discuss the shape optimisation of lattice-skin structures. We then introduce in Section 5 four examples of increasing complexity, with the first three involving the topology optimisation of lattice and lattice-skin structures and the last example combining topology and shape optimisation.
2 Governing equations and finite element discretisation
In this section, we briefly review the governing equations of lattice-skin structures and their finite element discretisation, for details refer to [11, 52]. It is assumed that displacements are small and the material behaviour is elastic. We model the shell as a Kirchhoff-Love thin-shell, the lattice as a pin-jointed truss structure and the two are coupled using Lagrange multipliers.
The displacement of the shell mid-surface is denoted with and the displacement of a lattice node, or joint, with the index with . The displacement of all lattice nodes is collected in the matrix , where is the total number of nodes. Henceforth, the superscripts and are used to distinguish between the thin-shell and the lattice variables. The total potential energy of the lattice-skin structure is composed of the thin-shell energy , the lattice energy , and the potential of the applied loads , i.e.
The lattice energy is comprised of the energies of the individual struts,
where is the internal energy density depending on the axial strain , is the cross-sectional area, and is the length of the strut with the index . It is assumed that the cross-sectional area of each strut is constant over its length. The internal energy density for an elastic material with Young’s modulus is given by
The potential energy of the displaced thin-shell takes the form
where is the membrane and
the bending strain energy densities depending on the membrane and bending strain tensorsand , respectively. For a thin-shell with the thickness and an elastic and isotropic material with Young’s modulus and Poisson’s ratio , the two energy densities are given by
where is a fourth-order geometry-dependent tensor. Finally, assuming, without loss of generality, that the only applied external loading is a distributed load , the potential of the applied loads takes the form
The displacements of the lattice nodes attached to the shell are required to be compatible with the shell displacement, see Figure 6. To enforce it, first we collect the indices of all lattice nodes attached to the shell in the set . Furthermore, as usual in geometrically exact shell theories [47, 9], the shell mid-surface is parameterised with curvilinear parametric coordinates . Hence, the compatibility of the lattice node displacements and the shell displacement requires
where is the displacement of the -th lattice node attached to the shell mid-surface at the point with the parametric coordinate . This compatibility constraint (7) is imposed with Lagrange multipliers via the augmented potential
The stationarity of this potential yields the following set of equilibrium and compatibility equations
where denotes the virtual displacement of the shell mid-surface.
We discretise the mid-surface of the thin-shell and its displacement with Catmull-Clark subdivision basis functions, which represent a generalisation of cubic tensor-product B-splines to unstructured meshes; see [11, 56] for details. After introducing the discretised mid-surface and the discretised displacements into the weak form (9), and subsequent numerical evaluation of integrals, we obtain the discrete system of equations
where and are the stiffness matrices of the thin-shell and the lattice; and
are the respective displacement vectors andis the external force vector of the shell; is an extraction matrix which multiplied with gives the displacements of the lattice nodes with , and is a matrix obtained by evaluating the subdivision basis functions at the parametric mid-surface coordinates corresponding to lattice nodes with . We write (10) more compactly as
3 Lattice topology optimisation
3.1 Problem statement and penalisation of cross-sectional areas
We optimise the topology of the lattice structure with a penalisation technique akin to the SIMP approach extensively used in topology optimisation of continuum structures . Intermediate cross-sectional areas are penalised to obtain an optimised lattice with only some of the struts removed and (almost) no changes to the cross-sectional areas of the remaining ones. The associated compliance topology optimisation problem considering the equilibrium equation (10) reads
where is the structural compliance of the lattice-skin structure, is the relative density vector yet to be defined; is the actual and the initial lattice material volume, and is the prescribed volume fraction of the lattice. Note that in (12a), according to (10), only the shell has an external loading and the shell stiffness matrix is independent of the relative density .
In our SIMP-like penalisation approach the element stiffness matrix of each strut is first expressed as
where is the original stiffness matrix of the strut with the index , and its ratio of the current and original cross-sectional areas and is defined as its relative density . Hence, the relative density may take the values , in line with (12d). To penalise the intermediate relative densities the strut element stiffness matrix is replaced with
with the penalised relative density
The penalisation parameter ensures that the struts with relative densities close to and are preferred; see Figure 7. In some applications it can be advantageous or desirable to have relative densities so that the optimised area can be larger than the original area . In such cases the definition (15) leads to a slope discontinuity at , which may affect the convergence of optimisation. To remedy this, the function can be replaced, for instance, with a Bézier curve with continuous derivatives at . In the examples presented in this paper, we consider only the case and apply this as a constraint during optimisation. In addition, we always apply the constraint to avoid the singularity of the global stiffness matrix.
3.2 Sensitivity analysis and filtering
For gradient-based optimisation the derivatives, or sensitivities, of the cost function and the constraints in (12) with respect to the cross-sectional areas are required. It is straightforward to show that the derivative of the cost function (12a) with respect to the penalised densities, taking into account the equilibrium equation (10), is given by
As known in topology optimisation of continuum structures, filtering techniques are needed to avoid checkerboard instabilities and excessive mesh dependency of the solution [45, 46]. Similar issues can be observed in lattice topology optimisation without filtering. Commonly, filtering is applied by convolving the computed sensitivities with a kernel, or filter. We propose a sensitivity filter for lattices consisting of unit cells, as illustrated in Figure 10. The support of the filter is defined by a prescribed filter radius describing a circle (in 2D) or a sphere (in 3D). A unit cell is considered within the filter support when its centroid lies within the support, see Figure (a)a. The filtered sensitivity of a unit cell is obtained by centering the filter at its centroid and calculating the weighted average of the sensitivities of the struts belonging to unit cells within the support. Hence, the filtered sensitivity of a unit cell is given by
where is the number of struts in the unit cells within the support, and is a weight according to a linearly decaying kernel function
where and are the coordinates of the centroid of the -th strut and the cell , respectively. The support of the filter should not be chosen smaller than a unit cell. Furthermore, the motivation for the division by the strut length in (18) is that the summation in the numerator can thus be interpreted as the weighted strain energy density, i.e. the strain energy per unit volume.
3.3 Lattice extraction and reconstruction
We assign the filtered sensitivity of a unit cell to each strut within the cell. The sensitivity of the struts belonging to several unit cells is obtained by averaging the respective unit cell sensitivities. To obtain the optimised lattice structure, struts with cross-sectional areas larger than a small user-defined threshold are extracted from the lattice, see Figure (a)a. Notice that struts unique to a unit cell are either all present or not in the optimised topology because they have the same sensitivities.
Subsequently, as illustrated in Figure (b)b the complete topology of a unit cell is recovered when there are any dangling diagonal struts in the optimised lattice. As the last step, additional struts are reintroduced in the concave void regions of the lattice, see Figure (c)c. The proposed extraction technique yields a rigid lattice with no mechanisms, or zero energy modes [39, 15]. In this work we consider only unit cells that are not mechanisms, or do not have zero energy modes, when considered individually.
4 Shape optimisation of the lattice-skin structure
4.1 Geometry parameterisation
We use the free-form deformation (FFD) technique to parameterise the overall shape of the lattice-skin structure. The structure with the physical domain is first immersed in a larger rectangular control prism , see Figure 15. An overbar denotes, here and in the following, domains and points related to the original, i.e. not optimised, lattice-skin structure. The control prism is discretised with a uniform grid consisting of control points, where . The coordinates of the control points are denoted with with the multi-index . Subsequently, the control prism is parameterised using trivariate Bernstein basis functions of degree . For the sake of simplicity in presentation, in the following we choose .
In shape optimisation the control point coordinates serve as design degrees of freedom. Each point with the coordinate in the original physical domain has a fixed parametric coordinate , which is straightforward to determine given that a uniform grid is used. Hence, the introduced Bernstein basis functions provide the following parameterisation of the physical domain
The overall shape of the physical domain is altered by displacing the control points . Applying a displacement to the original control points yields the displaced coordinates , and, in turn, this yields the deformed coordinates for points in the physical domain according to (20). For the sake of completeness, the univariate Bernstein polynomials in (20) are given by
4.2 Problem statement and sensitivity analysis
In shape optimisation we aim to find the optimal positions of the control points which minimise the compliance of the lattice-skin structure whilst satisfying all constraints. Hence, the shape optimisation problem can be formulated as
where and are the actual and the initial total lattice-skin material volume, respectively. The derivative, or sensitivity, of the compliance cost function with respect to the control point coordinates , taking into account (11), reads
where and are the shell and strut element stiffness matrices, and the summations are over the respective elements in the discretised structure. The derivative of the stiffness matrix of a shell element with the index is given by
where the summation is over the nodes of the -th shell element, and is the parametric control prism coordinate of the node with the index . Note that in subdivision surfaces the number of nodes of an element depends on the local connectivity of the mesh. The derivatives of the strut element stiffness matrices have a similar form and are given by
where the summation is now over the two nodes of the -th strut. Here, it is necessary to distinguish between the lattice nodes attached to the shell and those not. The derivatives of a lattice node with respect to the control points are calculated as
where the summation is over the nodes
of the shell element to which the lattice node is attached. Furthermore, it is assumed that the shell mid-surface is interpolated withwhere are the subdivision basis functions. In order to enforce the displacement coupling in shape optimisation, should be evaluated with the parameter in the attached shell element, while the positions of other lattice nodes are evaluated with the corresponding parameter in the control prism . In this manner the conformality between the lattice and the thin-shell is guaranteed to be maintained. For derivatives of the shell and lattice stiffness matrices with respect to nodal coordinates, in (24) and (25), we refer to [2, 53].
We proceed to demonstrate the application and efficacy of the proposed optimisation approach with four selected examples. In all the examples the compliance of the structure is the cost function and the material volume is prescribed. The first two examples are motivated by benchmark examples from continuum topology optimisation and concern the topology optimisation of lattice structures. In the third example, we optimise the lattice infill topology of a lattice-skin cantilever structure. The concurrent topology and shape optimisation of a lattice-skin roof structure is considered in the last example. We use in all examples the sequential quadratic programming (SQP) optimisation algorithm in the NLopt library . The original cross-sectional areas of the struts are set as upper limits in lattice topology optimisation.
5.1 Lattice cantilever
A 2D lattice cantilever of size is considered, see Figure (a)a. The left end of the lattice is fixed, while the remaining faces are free. A point load with magnitude 100 is applied at the mid-height on the right end. The periodic uniform lattice consists of square unit cells with two diagonals. The side length of the unit cells is , and the Young’s modulus of the material is . The total material volume of the original lattice is with each strut having the same cross-sectional area . A volume fraction of is prescribed for optimisation. To this end, prior to optimisation all cross-sectional areas are uniformly reduced to obtain a total material volume of . The cross-sectional areas of the struts are constrained not to exceed their initial values .
First, we investigate the choice of the penalisation factor . Three different penalisation factors are considered, in turn. The radius of the filter is , i.e. it is twice the unit cell side length. The optimised cross-sectional areas of the struts are depicted in Figure (b)b, and the frequency of the cross-sectional areas in Figure (c)c. As can be seen, a too-small penalisation factor leads to many struts with intermediate cross-sectional areas. In addition, a too-small penalisation factor may lead to checkerboards known from continuum optimisation. Clearly, these issues can be alleviated by choosing a relatively large penalisation factor. Although the optimised structures with a lower penalisation achieve a smaller compliance, as indicated in the caption of Figure (b)b, struts with deficiently small cross-sectional areas may buckle and can be impossible to manufacture.
Next, we consider three different lattices with the same total material volume, but with three different unit cell sizes , and . The penalisation factor is chosen as , and the filter radius is twice the unit cell size. The optimised lattices shown in Figure 23 are obtained by removing struts with cross-sectional areas less than and recovering the complete topology of unit cells which have any remaining diagonal struts, cf. Figure 14. As visually apparent the optimised topology and geometry remain the same irrespective of the unit cell size. Furthermore, as can be analytically explained the compliance becomes larger with decreasing unit cell size .
5.2 Lattice MBB beam
The simply supported MBB beam is a widely studied benchmark example in continuum topology optimisation. We consider an equivalent lattice structure as depicted in Figure (a)a. The beam has the size and is subjected to a vertical point load of at the midpoint of its top face. To begin with, we choose a periodic uniform lattice with unit cells of size and each having two diagonals. The Young’s modulus of the material is and all the struts have the same cross-sectional area of giving a total material volume of . The compliance of the non-optimised structure is .
A volume fraction is prescribed for optimisation. Furthermore, the penalisation factor is and the filter radius is (i.e. twice the unit cell size). The optimised cross-sectional areas of the struts are depicted in Figure (b)b. The final structure in Figure (c)c is obtained by removing struts with cross-sectional areas less than and recovering the complete topology of unit cells which have any remaining diagonal struts. The structural compliance of the optimised lattice is , which is a reduction by .
To demonstrate the application of the proposed approach to periodic non-uniform lattices, we consider the lattice with non-uniform unit cell sizes shown in Figure (a)a. The cells are designed to be denser in the mid-span where the axial forces are largest. Again, a volume fraction of is chosen for optimisation. The structural compliance of the initial lattice is , which is reduced to for the optimised structure in Figure (c)c. Compared to the uniform lattice structure, the non-uniform structure is stiffer due to the purposefully chosen distribution of unit cell sizes, indicating the potential in optimising the lattice layout (geometry) to further reduce the compliance.
Topology optimisation of the lattice MBB structure with a non-uniform distribution of unit cell sizes.
5.3 Lattice-infilled cantilever
We consider next the cantilever shown in Figure (a)a comprised of a lattice and exterior thin-shell face sheets. The left end is clamped while the remaining faces are free. At the centre of the right face a distributed load with a total magnitude of over an area of is applied. The cantilever has the size . The periodic uniform lattice consists of BCC (body-centred cubic) unit cells of side length . The Young’s modulus is and the Poisson’s ratio is . The total material volume is 40. The volume fraction is chosen as , the penalisation factor as and the filter radius as . In order to examine the effect of the thin-shell thickness on the structural performance, three different thicknesses are considered. The total material volume is fixed so that the respective strut diameters are , and . During the topology optimisation, the material volume ratio between the lattice and the thin-shell are preserved, i.e. the thickness of the thin-shells is fixed.
The lattice topology optimisation results for the three different lattice-to-shell volume ratios are shown in Figures (b)b, (c)c and (d)d. Clearly, the optimal lattice topology depends on . When the lattice volume is relatively large, the optimised structure resembles the ones obtained from lattice-only, c.f. Section 5.1, and continuum topology optimisation. However, with decreasing lattice volume the lattice-like topology starts to disappear, as can be seen in Figures (c)c and (d)d. This can be structurally explained as follows: when the shell thickness is very small, the shear rigidity in the height direction is provided by the lattice, whereas when the shell becomes thicker, the shear rigidity is increasingly provided by the shell faces. This leads to a concentration of the struts at the top and bottom regions of the cantilever where the axial stresses due to bending are large.
It is worth emphasising that as the lattice volume decreases, the structural compliance of the optimised cantilever decreases as well, indicating that the lattice structure is not optimal compared with a solid-only structure in terms of compliance . Nonetheless, a lattice can be optimal when in addition to compliance other design or performance criteria have to be taken into account, such as the buckling of the face sheets.
5.4 Pentagon lattice-skin roof
As the last example we consider a lattice-skin roof structure to demonstrate the application of the proposed concurrent shape and lattice topology optimisation approach. The initial geometry of the roof structure is obtained starting from a pentagon-shaped flat shell, i.e. plate, supported by a single layer of lattice consisting of BCC periodic unit cells, see Figure (a)a. The radius of the circumcircle of the regular pentagon is and the shell thickness is . The lattice has a height of and the struts have a diameter of . The Young’s modulus and the Poisson’s ratio are and . A uniform pressure with a magnitude of is applied on the shell mid-surface. The roof is supported at the middle of the five pentagon edges.
Prior to optimisation, we obtain the curved roof structure in Figure (b)b by scaling the deflection of the initially flat roof under a uniform pressure loading. This form-finding step is motivated by the hanging chain or cloth models commonly used in architectural design. The so-obtained curved lattice is guaranteed to be conformal to the curved shell mid-surface because of the coupling between the lattice and shell nodes. The compliance of the curved but not yet optimised roof is .
First, we optimise the shape of the lattice-skin roof after parameterising its geometry with the FFD technique presented in Section 4. During iterative geometry updating, the positions of the roof supports are fixed in space by placing FFD control points at the roof supports and choosing them as fixed. The final shape optimised roof is shown in Figure (c)c. The structural compliance is reduced to , which gives a reduction of compared with the initial curved roof. Figure (d)d shows the displacement of the shape optimised roof under a uniform pressure loading.
Next, the topology of the supporting lattice is optimised to reduce further the material usage without significantly compromising structural stiffness. The shape optimised roof still contains a large number of underutilised struts many of which can be removed. As an ancillary benefit, the removal of struts may improve the natural lighting conditions in the space under the roof, which is often an important architectural design consideration. For lattice topology optimisation a volume fraction of is prescribed. The penalisation factor is , and the filter radius is . In the first step the cross-sectional areas of all struts are uniformly reduced, as shown in Figure (a)a, without changing topology. As a consequence of the reduced lattice volume, the compliance jumps from to . The final shape and topology optimised roof is shown in Figure (b)b, which has the compliance . The compliance increases slightly by compared with the shape optimised roof, while the lattice volume is reduced by , which indicates that it is possible to remove a large number of struts without compromising the stiffness greatly. Figure (c)c shows the displacement of the final shape and topology optimised roof under uniform pressure loading.
We introduced a novel concurrent shape and infill topology optimisation approach that fully takes into account the coupling between the pin-jointed lattice and the thin-shell skin. The modelling of the lattice as a pin-jointed truss allows us to relax the uniformity and scale separation assumptions underlying homogenisation theories. The proposed SIMP-like method for penalising struts with intermediate cross-sectional areas yields a black and white design, which is essential for manufacturability and satisfying additional constraints, like strut buckling. In shape optimisation, we use the free-form deformation technique because it allows us to seamlessly parameterise the shape of the entire lattice-skin structure. The presented examples demonstrate the robustness and efficiency of the proposed approach in minimising the compliance of lattice and lattice-skin structures for a given volume of material.
The presented lattice-skin optimisation approach can be extended in several ways. In this paper, we only optimised the compliance for a prescribed volume and a single load case. In practice, there are many more load cases, competing cost functions and constraints that have to be taken into account. We expect that most of these can be included without significant modifications to the presented approach . Furthermore, we modelled the lattice as a pin-jointed truss which is sufficient for engineered stretch-dominated lattices with appealing mechanical properties. Alternatively, to model bending-dominated lattices or to take into account secondary bending effects, the struts can be modelled as beams that are rigidly connected at the nodes . This will increase the number of lattice degrees of freedom by a factor of two and will require the coupling of the rotations of the joints attached to the shell and the tangent plane rotations of the shell mid-surface. Finally, the manufacturability of the optimised geometries is a crucial consideration in practice [32, 31]. Although we did not consider any manufacturability constraints, a discrete lattice model should greatly aid their imposition.
-  (2007) On simultaneous optimization of truss geometry and topology. Structural and Multidisciplinary Optimization 33, pp. 285–304. Cited by: §1.
-  (2018) Isogeometric shape optimisation of shell structures using multiresolution subdivision surfaces. Computer-Aided Design 95, pp. 62–71. Cited by: §1, §4.2.
-  (1994) Optimization methods for truss geometry and topology design. Structural Optimization 7, pp. 141–159. Cited by: §1.
-  (1988) Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering 71, pp. 197–224. Cited by: §1.
-  (1999) Material interpolation schemes in topology optimization. Archive of Applied Mechanics 69, pp. 635–654. Cited by: §1, §3.1.
-  (2014) A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Structural and Multidisciplinary Optimization 49, pp. 873–895. Cited by: §1.
-  (1984) Shape optimal design using b-splines. Computer Methods in Applied Mechanics and Engineering 44, pp. 247–267. Cited by: §1.
-  (1981) Buckling of shells - pitfall for designers. AIAA journal 19, pp. 1183–1226. Cited by: §1.
-  (2005) An introduction to differential geometry with applications to elasticity. Journal of Elasticity 78, pp. 1–215. Cited by: §2.
-  (2011) Subdivision shells with exact boundary control and non-manifold geometry. International Journal for Numerical Methods in Engineering 88, pp. 897–923. Cited by: §1.
-  (2000) Subdivision surfaces: A new paradigm for thin-shell finite-element analysis. International Journal for Numerical Methods in Engineering 47, pp. 2039–2072. Cited by: §1, §2, §2.
-  (2002) Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision. Computer-Aided Design 34, pp. 137–148. Cited by: §1.
-  (2015) Topology optimization of coated structures and material interface problems. Computer Methods in Applied Mechanics and Engineering 290, pp. 524–541. Cited by: §1.
-  (2017) Topology optimization of 3d shell structures with porous infill. Acta Mechanica Sinica 33, pp. 778–791. Cited by: §1.
-  (2001) Foam topology: bending versus stretching dominated architectures. Acta Materialia 49, pp. 1035–1040. Cited by: §3.3.
-  (2001) Effective properties of the octet-truss lattice material. Journal of the Mechanics and Physics of Solids 49, pp. 1747–1769. Cited by: §1.
-  (2010) Micro-architectured materials: past, present and future. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 466, pp. 2495–2516. Cited by: §1.
-  (2014) Additive manufacturing technologies. Vol. 17, Springer. Cited by: §1.
-  (1999) Cellular solids: structure and properties. Cambridge University Press. Cited by: §1.
-  (2003) Layout optimization of large-scale pin-jointed frames. Engineering Computations 20, pp. 1044–1064. Cited by: §1.
-  (2018) Homogenization-based topology optimization for high-resolution manufacturable microstructures. International Journal for Numerical Methods in Engineering 113, pp. 1148–1163. Cited by: §1.
-  (2009) Topology optimization of trusses by growing ground structure method. Structural and Multidisciplinary Optimization 37, pp. 377–393. Cited by: §1.
-  (2014) An adaptive geometry parametrization for aerodynamic shape optimization. Optimization and Engineering 15, pp. 69–91. Cited by: §1.
-  (2019) The embedded isogeometric kirchhoff–love shell: from design to shape optimization of non-conforming stiffened multipatch structures. Computer Methods in Applied Mechanics and Engineering 349, pp. 774–797. Cited by: §1.
-  (2005) Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, pp. 4135–4195. Cited by: §1.
-  (1982) Three-dimensional shape optimization. International Journal for Numerical Methods in Engineering 18, pp. 661–673. Cited by: §1.
-  (2020) The nlopt nonlinear-optimization package. External Links: Cited by: §5.
-  (2014) Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting. Computer Methods in Applied Mechanics and Engineering 274, pp. 148–167. Cited by: §1.
-  (1989) Optimal topologies of structures. Applied Mechanics Reviews 42, pp. 223–239. Cited by: §1.
-  (2011) A gradient-based, parameter-free approach to shape optimization. Computer Methods in Applied Mechanics and Engineering 200, pp. 985–996. Cited by: §1.
-  (2018) Current and future trends in topology optimization for additive manufacturing. Structural and Multidisciplinary Optimization 57, pp. 2457–2483. Cited by: §6.
-  (2016) A survey of manufacturing oriented topology optimization methods. Advances in Engineering Software 100, pp. 161–175. Cited by: §6.
-  (2012) Shear-flexible subdivision shells. International Journal for Numerical Methods in Engineering 90, pp. 1549–1577. Cited by: §1.
-  (2007) Growth method for size, topology, and geometry optimization of truss structures. Structural and Multidisciplinary Optimization 33, pp. 13–26. Cited by: §1.
-  (2011) Optimization of finite dimensional structures. CRC Press. Cited by: §1, §6.
-  (2018) Strategies for functionally graded lattice structures derived using topology optimisation for additive manufacturing. Additive Manufacturing 19, pp. 81–94. Cited by: §1.
-  (2008) A post-treatment of the homogenization method for shape optimization. SIAM Journal on Control and Optimization 47, pp. 1380–1398. Cited by: §1.
-  (1999) Scale effects on the elastic behavior of periodic and hierarchical two-dimensional composites. Journal of the Mechanics and Physics of Solids 47, pp. 1509–1542. Cited by: §5.1.
-  (1986) Matrix analysis of statically and kinematically indeterminate frameworks. International Journal of Solids and Structures 22, pp. 409–428. Cited by: §1, §3.3.
-  (2004) Shell structures—a sensitive interrelation between physics and numerics. International Journal For Numerical Methods In Engineering 60, pp. 381–427. Cited by: §1.
-  (2016) An efficient and scalable approach for generating topologically optimized cellular structures for additive manufacturing. Additive Manufacturing 12, pp. 296–304. Cited by: §1.
-  (2016) Architected cellular materials. Annual Review of Materials Research 46, pp. 187–210. Cited by: §1.
-  (1986) Free-form deformation of solid geometric models. In Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques, pp. 151–160. Cited by: §1.
-  (2016) On the (non-) optimality of michell structures. Structural and Multidisciplinary Optimization 54, pp. 361–373. Cited by: §5.3.
-  (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization 16, pp. 68–75. Cited by: §3.2.
-  (2007) Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization 33, pp. 401–424. Cited by: §3.2.
-  (1989) On a stress resultant geometrically exact shell model. part i: formulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering 72, pp. 267–304. Cited by: §2.
-  (2013) Cost-effective printing of 3d objects with skin-frame structures. ACM Transactions on Graphics 32, pp. 177:1–177:10. Cited by: §1.
-  (2017) Concurrent design with connectable graded microstructures. Computer Methods in Applied Mechanics and Engineering 317, pp. 84–101. Cited by: §1.
-  (2018) A level set method for shape and topology optimization of coated structures. Computer Methods in Applied Mechanics and Engineering 329, pp. 553–574. Cited by: §1.
-  (2017) Minimum compliance topology optimization of shell–infill composites for additive manufacturing. Computer Methods in Applied Mechanics and Engineering 326, pp. 358–375. Cited by: §1.
-  (2019) Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures. Computer Methods in Applied Mechanics and Engineering 351, pp. 928–950. Cited by: §1, §2.
-  (2020) Topologically robust CAD model generation for structural optimisation. Computer Methods in Applied Mechanics and Engineering 369, pp. 113102. Cited by: §4.2, §6.
-  (2014) GRAND—ground structure based topology optimization for arbitrary 2d domains using matlab. Structural and Multidisciplinary Optimization 50, pp. 861–882. Cited by: §1.
-  (2016) Bridging topology optimization and additive manufacturing. Structural and Multidisciplinary Optimization 53, pp. 175–192. Cited by: §1.
-  (2018) Subdivision surfaces with isogeometric analysis adapted refinement weights. Computer-Aided Design 102, pp. 104–114. Cited by: §2.