Optimal design of fibre reinforced membrane structures

A finite element membrane shell model was recently derived by Hansbo and Larson (2014) using tangential differential calculus, meaning that the problem is set in a Cartesian three dimensional space as opposed to a parametric plane, thereby generalizing the classical flat facet element shell model to higher order elements. The present study further extends this membrane model by allowing for non-isotropic materials. In particular, one–dimensional fibers are added to a base material, modeling, e.g., the reinforcements seen in modern racing boat sails. The plane stress property, as well as the membrane property of complete out-of-plane shear flexibility, is shown to be exact consequences of certain material parameter selections for a three-dimensional transversely isotropic base material. This together with a displacement assumption reduces the three dimensional model to the surface model. Based on this finite element model we formulate a design problem where we seek to find the best fiber reinforcements of the membrane, meaning that we find the stiffest structure by both rotation and sizing of the fibers. The formulation consists of two minimization statements. Since these two statements relate to rotation and sizing of the fibers, respectively, such a formulation ties directly to the sequential iterative treatment suggested for similar problems previously (Bendsøe and Sigmund 2002). The optimal rotation is found by identifying the material as a so-called low shear material, implying that the optimal orthotropic principal directions coincides with the principal stress directions (Pedersen 1989, 1991), while the optimal thickness distribution is found by a classical optimality criteria iteration formula. On a more general level the approach is reminiscent of so called block coordinate descent methods (Bertsekas 1999; Beck and Tetruashvili 2013).

We define the membrane forces (per unit length) as Stationarity of the potential energy gives the following principle of virtual work: for all kinematically admissible fields v. Such fields will generally be restricted in the tangential direction on a subset of ∂Σ. We will assume that loading on the membrane can be written as where f is a force per area over Σ, and p is a force per unit length over the part S of ∂Σ where the displacement is not prescribed. Using now Lemma 2.1 of Gurtin and Murdoch (1975), i.e., an integral theorem for surfaces, we obtain the equilibrium equations where div_Σ is the surface divergence, and ν is a unit vector of ∂Σ, tangential to Σ. Since M ν will also be a vector tangent to Σ we conclude that p can have no component perpendicular to the surface.

From now on we will consider the special case of an orthotropic material consisting of two orthogonal families of fibers, consisting of the same material, i.e., α ₁ = α ₂ = α. We use the notation s =s ₁ and s ^⊥ =s ₂.

The orientation of the fibers in the tangent plane of the membrane, i.e., s and s ^⊥, can be defined by an angle 𝜃 belonging to This angle will be a design variable in the optimal design problem. Other such design variables are t ₁ and t ₂, i.e., the fiber contents in the two orthogonal directions. The field t = (t ₁,t ₂) belongs to the set where u n d e r l i n e t _α and o v e r l i n e t _α are non-negative upper and lower bounds and V is a limit for the total amount of material that can be used for the fibers.

The potential energy is seen as a function where V is the set of kinematically admissible displacements. Minimizing π with respect to the first argument gives the equilibrium displacement as a function of the design variables, i.e., u = u(t,𝜃). As a measure of stiffness we use the so called compliance Our design goal is to find a design that minimizes the compliance. We choose to split this into two parts as follows: find t ^∗ ∈ T and 𝜃 ^∗∈ Θ such that The splitting into two minimization statements is partly motivated by the numerical treatment, where the two subproblems are solved in sequence, reminiscent of a block coordinate descent method, Bertsekas (1999) and Beck and Tetruashvili (2013). However, it is also motivated by the fact that each of the two problems are well-posed in the sense that existence of solutions can be proved, as discussed below.

The second sub-problem of \((\mathbb {P})\), i.e., finding an optimal orientation for an orthotropic material, has been extensively discussed by Pedersen (1989, 1991) and Hammer (1999), where it is treated in its present form, using 𝜃 as variable. However, the problem can also be rewritten in terms of so-called lamination parameters, see Hammer et al. (1997) and Bendsøe and Sigmund (2002). Such a rewriting gives an objective that is linear, making a proof of existence of solutions possible and thereby indicating that regularization, e.g., by filters, is not necessary. However, checkerboard-like patterns were found in the finite element scheme used in Thomsen and Olhoff (1990) and Thomsen (1991) for a particular choice of material, interpreted as having one fiber direction only. Such anomalies have not been seen in our calculations and following Pedersen (1989, 1991) the second sub-problem of \((\mathbb {P})\) is solved locally, i.e., the orientation of the material is determined by the local stress state only, and in particular the orientation of principal stresses and strains. Due to the plane stress assumption there are only two possibly non-zero principal components of the stress tensor σ, denoted σ _I and σ _{I I} , such that |σ _I |≥|σ _{I I} |. The corresponding principal directions (eigenvectors) are tangent to the membrane plane. Obviously, these facts also holds for the principal components of M, i.e., M _I and M _{I I} , such that |M _I |≥|M _{I I} |. For a so-called low shear orthotropic material, the solution 𝜃 ^∗ of the second sub-problem of \((\mathbb {P})\) represents an orientation where the orthotropic principal directions coincide with the principal stress or membrane force directions, which are also the principal strain directions. Moreover, the orthotropic principal direction having the highest stiffness should be in the direction corresponding to σ _I and M _I . In the Appendix we show that the particular orthotropic material defined above, having two families of fibers in orthogonal directions s and s ^⊥, is indeed a low shear material and, therefore, the optimal directions of s and s ^⊥ are in the directions of principal stress. Moreover, if t ₁ > t ₂ then s is in the direction of σ _I .

The first sub-problem of \((\mathbb {P})\) is a classical stiffness optimization problem, albeit having two design fields, one for each fiber orientation. This is a convex problem that can be proved well-posed (see, e.g., Petersson 1996) and regularization by filters is not needed. It can be solved by satisfying the optimality conditions, leading to a fixed point iteration formula. The surface elasticity tensor \(mathbb{S}^{\text { memb}}=t\,\mathbb {E}^{\text { memb}}\) is regarded as a function of the design, i.e., \(mathbb{S}^{\text { memb}}=\mathbb {S}^{\text { memb}}(\boldsymbol {t},\theta )\). The optimality conditions of the first sub-problem of \((\mathbb {P})\) become (Bendsøe and Sigmund 2002; Christensen and Klarbring 2009):

where Λ, \(lambda_{\alpha }^{+}\) and \(lambda_{\alpha }^{-}\) are Lagrangian multipliers, t ∈ T and u = u(t,𝜃) is the displacement solution, i.e., the minimum field with respect to v of π(v,t,𝜃).

Note that

Concerning existence of solutions to the full problem \((\mathbb {P})\), this, as well as proof of convergence of the iterative treatment, seems to be open questions. However, experience in similar problems, e.g., in Bendsøe and Sigmund (2002) and Thomsen and Olhoff (1990), indicates that the approach is viable.

For the numerical treatment of \((\mathbb {P})\) we need to introduce a discrete approximation. The discretization of the state problem, i.e., the problem of finding the minimum displacement u ∈ V of the potential energy π for a given design 𝜃 ∈ Θ and t ∈ T, follows Hansbo and Larson (2014). This implies introducing a triangulation of Σ resulting in a discrete surface, with corresponding discrete normal vector field and projections. The displacement field is approximated using the same triangulation but is possibly of different order.

In addition to the approximation of the state problem we also need to approximate the design fields t ∈ T and 𝜃 ∈ Θ. This is achieved by using point values: these are denoted t _i = (t _1i ,t _2i ) and 𝜃 _i for point i. In particularly, we use superconvergence points of the finite elements (Barlow 1976). Such a discretization means that (14) and (16) are imposed at these evaluation points and the integral in (15) is replaced by a sum.

Let be the left hand side of (14) evaluated at point i and for a displacement field u ^k . Also, let \(B^{k}_{\alpha i}=({\Lambda }^{k})^{-1}A^{k}_{\alpha i}\) where Λ ^k is a current iterate of the Lagrangian multiplier Λ. For a given displacement iterate u ^k and rotation 𝜃 ^k the following fixed point iteration formula is suggested by the optimality conditions (14) through (16): where u n d e r l i n e t _{α i} and o v e r l i n e t _{α i} are point values of the upper and lower bounds and 0 < η ≤ 1 is a damping coefficient.

The following algorithmic steps, the convergence of which gives satisfaction of a discrete version of the optimality conditions of \((\mathbb {P})\), are now suggested: Steps 1 and 2 can be iterated several times before continuing with calculation of fiber directions in Step 3. In fact, in the examples the fixed point iteration (17), for newly calculated displacement u ^k , is repeated until convergence before continuing with the fiber directions in Step 3.

Note that step 3 assumes distinct principal stresses. Numerically coalescence of such stresses occur with close to zero probability but may show up as non-convergence issues. For statically determined structures, i.e., when M is uniquely determined by (12) and (13), this may be of particular concern. For such cases that have distinct principal stresses, step 3 above needs to be performed only once since these principal stresses are independent of t. Such problems essentially become convex since the first part of \((\mathbb {P})\) is a convex problem. The first problem of Section 6 is statically determinate but has not everywhere distinct principal stresses.

Clearly other algorithms than the optimality criteria algorithm are available for solving the first part of problem \((\mathbb {P})\). General purpose sequential convex approximation methods (Christensen and Klarbring 2009) like the Method of Moving Asymptotes (MMA) (Svanberg 1987) could be directly applied. A variable thickness sheet problem, that has the same structure and convexity property as the first part of \((\mathbb {P})\) was solved by MMA in Christensen and Klarbring (2009), generally needing more iterations than the optimality criteria algorithm. However, since convergence properties could be sensitive to parameter values it is hard to draw general conclusions based on this experience. Moreover, the optimality criteria algorithm can in fact itself be seen as a particular first order sequential convex approximation method (Christensen and Klarbring 2009) and, thereby, shows similar properties as MMA. In Thomsen and Olhoff (1990) and Thomsen (1991) the first part of \((\mathbb {P})\) was solved by the CONLIN method, see, e.g., Christensen and Klarbring (2009), which is yet another first order method. The same range of number of iterations as in this paper were needed for convergence. Again indicating that essentially any first order method shows similar behaviour.

A distinctively different algorithmic treatment of \((\mathbb {P})\) is to solve both types of variables - rotation and sizing - simultaneously by a general purpose method. However, such an approach would not utilize that the second part of \((\mathbb {P})\) has a known solution in terms of given stresses, and, in particularly, would not use the special property of statically determinate problems. Moreover, an explicit parametrization of the rotation is needed for calculation of sensitivities. Such a parametrization would involve a cyclic variable, which together with non-convexity may make a simultaneous approach more likely to end up in local minima.

The classical facet approach to membrane shells was recently extended to curved elements by Hansbo and Larson (2014). Here we make a further extension by showing how orthotropic material, of fiber type, can be treated in a similar way, partly inspired by exact plate theory of Nardinocchi and Podio-Guidugli (1994). Based on this orthotropic membrane shell theory we formulate a stiffness design problem, where we seek an optimal structure by both rotation and sizing of reinforcing fibers. The two design variables - representing rotation and sizing - naturally split the formulation into two minimum statements, which suggests a sequential numerical treatment, previously used for similar problems (Bendsøe and Sigmund 2002). This type of formulation also makes clear the distinct character of statically determined problems, which occur for large classes of membrane shells (Ciarlet 2000). For such problems, the material independent stress state implies that the two minimization statements of \((\mathbb {P})\) become decoupled, and the full problem then means solving a convex sizing problem and separated local orientation problems, each with known analytical global minima.

The approach presented in this paper has several intriguing extensions, that would be important for applications such as the design of racing boat sails. Inclusion of pre-stress and wrinkling states related to negative stresses are examples of this. Extension to large deformations, based on the model of Hansbo et al. (2015), should also be of clear interest.

It may also be noted that, while the large design freedom allowed when optimizing orientation and thickness independently at each point may seem unrealistic from a manufacturability point of view, restrictions involving patches of equal designs can easily be added to the formulation. However, new manufacturing methods, like 3D printing, constantly reduces the need for such constraints.