StrucOpt manuscript No. èwill be inserted by the editorè Detecting the stress æelds in an optimal structure II: Three-dimensional case A. Cherkaev and

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1 StrucOpt manuscript No. èwill be inserted by the editorè Detecting the stress æelds in an optimal structure II: Three-dimensional case A. Cherkaev and _I. Kíucíuk Abstract This paper is the second part of the investigation on stress æelds in an optimal elastic structure. In the ærst part of this research, Cherkaev and Kíucíuk èè, we derived the necessary conditions for the stress æeld in optimal two-dimensional elastic structures and introduced a method to check if a structure is optimal. In this paper, we turn our attention to conditions of the stress æeld in optimal three-dimensional elastic structures. We restate the necessary conditions for minimization of the stress energy in three-dimensional elasticity. We also show that the conditions are realized in optimal microstructres. Introduction The structures for optimal three-dimensional elasticity were introduced and investigated in the papers: Gibiansky and Cherkaev è987è Lipton and Diaz è99è Cherkaev and Palais è997è Allaire et al. è997è Olhoæ et al. è998è Kíucíuk èè. The authors used the sufæcient conditions ètranslation methodè to compute the lower bound of the energy and demonstrated that the guessed structure correspond to this bound this demonstration proved simultaneously the optimality of structures and bounds. Here we investigate the æelds inside of optimal structures using classical technique of the calculus of variations. The method of structural variation was introduced in books Lurie è97, 99è we use a version of it developed in a book Cherkaev èè. The stress æelds within an optimally designed elastic structure satisæes certain conditions. These conditions show that the homogenized constitutive equations are on Received: August, 998 A. Cherkaev and _I. Kíucíuk Department of Mathematics, University of Utah, Salt Lake City, Utah, USA cherk@math.utah.edu and kucuk@math.utah.edu the boundary of ellipticity. Accordingly, the homogenized energy of an optimally designed elastic structure is on the boundary of quasiconvexity èsee, for example, Cherkaev èèè. Here, we derive and analyze the pointwise æelds in optimal structures. Most of the calculations are performed using MAPLE. If the derived formulas are too bulky, we do not display them, here instead we refer to ægures and the algorithm given in the Appendix. Detailed calculations can be found in Kíucíuk èè. Formulation of the Problem Geometry Consider a domain æ that is divided into two subdomains æ and æ. Suppose that the subdomains æ i are occupied by isotropic materials with bulk and shear moduli of i and i, respectively. Suppose also that the volumes of æ and æ are æxed: Z dæ = M æ èxè is the characteristic function deæned as follows èxè = if x æ if x æ : èè Finally, assume that æ and æ are the cost of Material and Material, respectively. Elasticity Consider the elastic equilibrium in the domain æ, assuming the absence of body forces a load is applied from the boundary, T of æ. A linearly elastic structure at equilibrium satisæes elasticity equations ræèxè = æ = èru +èruèt è æèxè = Sèxèèxè èè èxè, and æèxè are three-by-three stress and strain tensors respectively, is the gradient,

2 and Sèxè is the compliance tensor, carrying information about the material properties and associating two æelds. The energy of the elastic equilibrium is equal to Gè è = èxè :Sèxè :èxè èè second-rank stress tensors èxè belong to the set F s èxè of statically admissible stress æelds F s èxè = fèxè jræèxè =in æ èxènèxè =tèxè on, T g: èè Here, tèxè isgiven surface tractions nèxè is the normal vector to the surface and, T is the surface the traction is applied. A fourth-rank material compliance tensor Sèxè belongs to the set of admissible compliance tensors, S ad Sèxè =èxès è è+è, èxèès è è èè i and i are shear and bulk moduli of the i-th material in the domain æ. The equilibrium of the structure corresponds to the principle of minimum of total stress energy equations èè are the Euler-Lagrange equation for the variational problem Eèè = min èxèf s èxè Z Gè èdæ æ èè a solution èxè delivers the the minimum of the total stress energy èor maximum stiæness problemè. Optimal design Consider now a problem of optimal structure: Find èxè such that J = min Wèè èxè Wèè =ëeèè+æ +æ è,èë è7è It is expected that a solution to the optimization problem è7è includes æne-scale oscillations of the control Sèxè èsee, for example, Cherkaev èèè in other words, the optimally designed structure is a composite or a limit of rapidly oscillating sequences of the original controls. The optimal distribution of the two materials can be described by a fastly oscillating sequence between the domains æ i.to deal with the fastly oscillating solutions, relaxation techniques are developed. The relaxation technique essentially replaces the original optimization problem with another one that has a classical solution. Here we investigate the æelds in the pure materials regardless of how wiggle is the line dividing the regions of them. Therefore, we do not use homogenization until ænal interpretation of results.. Notations for Calculations of Three-dimensional Elastic Composite Calculations in a three-dimensional problem can be tedious unless appropriate notations are introduced. Therefore, the following basis is introduced to transform fourthand second-rank tensors into six-by-six matrices or sixby-one vectors, respectably. b èè = i æ i b èè = p èi æ i + i æ i è b èè = i æ i b èè = p èi æ i + i æ i è b èè = i æ i b èè = p èi æ i + i æ i è è8è i =èè T i =èè T and i =èè T : è9è are æxed reference Cartesian coordinates and dyadic product of two vectors, æ, is deæned as G = a æ b = a æ b T or èè G ij = a i b j which is a second-rank tensor. Similarly, the dyadic product of m th -rank tensor and k th -rank tensor results in corresponding to èm+kè th -rank tensor. The transformation rules D ij = b èiè ææ S ææb èjè i j = : èè allow to express any stress tensor as a six-dimensional vector in the basis è8è: =è p p p è T : èè Together with the notation èè, we shall use the following simpler notation in some calculations hereafter s =ès s s s s s è T : èè Similarly, any fourth-rank isotropic compliance tensor S can be represented as six-by-six matrix D i = d d d d d d d d d d d d 7 d = i + i d =, i, i and d = : 9 i i 8 i i i èè

3 The matrix D i is nondiagonal. Although we used a new notation D for a transformed compliance tensor, S will also be used for transformed compliance tensors hereafter in accordance with the usual convention. Variations. The Scheme of the Weierstrass Test A necessary condition of optimality, namely, the Weierstrass test is used to investigate the optimal design. This test deals with the increment of the functional caused by the special structural variation. To perform the variation, we implant inænitesimal ellipsoidal inclusions ælled with an admissible material S n at a proximity of a point x in the domain æ h that is occupied by a host material S h. We compute the increment: The diæerence between the cost of the problem in two conægurations with and without inclusion. If the examined structure is optimal, then the increment is nonnegative. The increment depends on the shape of the region of variation and must stay nonnegative for all inclusions therefore the strongest condition corresponds to such a shape that the increment reaches its minimal value which must be also nonnegative. If this condition is violated, then the cost could be reduced by a variation, and the structure fails the test. The Weierstrass test was suggested in the described form by Lurie in the book Lurie è97è.. Variation of Properties: Three Dimensions To calculate the variation of properties, add a quasiperiodic dilute composite of third-rank laminates to material S h in a neighborhood of the point x in accord with Cherkaev èè. In these laminates, the inclusions are made of material S n, and the envelope is made of material S h. In other words, we construct a third-rank laminate from the isotropic host medium contained in the envelope. This matrix laminate composite of third-rank is characterized by its eæective tensor S æ given in Gibiansky and Cherkaev è997aè, èsee also Francfort and Murat è98è and Cherkaev èèè as S æ èmè =S h + m, ès n + S h è, +è, mèn rd èæè æ, èè m is the volume fraction of the nuclei material the matrix S i in the basis of è8è is given by èè and matrix N rd determines the geometry of the laminate: N th èæè = X i= æ i N i X i= æ i = æ i : èè N i = p i èp T i S p i è, p T i : è7è The matrix projector p i maps the stress vector èè into its discontinuous part it depends on the normal n to the layers in the structure p i = p i ènè: è8è Projection matrix p i in N i 's of èè are given by èsee Gibiansky and Cherkaev è997aèè: p = 7 p = 7 and p = 7 : è9è In è9è, the projection matrices point to the discontinuous components of stress æelds in èè. The lamination directions n k in the structure è8è are directed along i k. Substituting èè, èè, and è9è into èè and using the following notations for æ's: in èè result in N rd = æ = æ æ = æ and æ =, æ, æ è, æèe æ e æe æ e è, æèe æe æe æe èæ + æèe æ e æe æe 7 èè æ =, èæ + æè e = è h+ h è h h + h e = è h, h è h h + h and e = h. The eigenvectors of N rd are computed in the basis è8è that coincide with the directions of lamination in R the inner parameters æ i are responsible for relative elongation èintensitiesè of the inclusions. Dilute inclusions The increment æs caused by the array of inænitely dilute nuclei with material S n and inænitesimal volume fraction æmèxè isgiven by Oèæè if kx, x k éæ æmèxè = æm C if kx, èæè èè x kæ: :

4 similarly to the two-dimensional case. The eæective property of this composite becomes S æ èm + æmè it is calculated by Taylor's expansion S æ èm + æmè =S æ èmè+æm d dm S æèmè+oèæmè: If we substitute the value of S æ èmè from èè into the last equation and compute. Increment The increment of the functional consists of the direct cost caused by change of quantities of the used materials and the increment of energy. The ærst term is easy to compute: Replacing the host material S h èwith the speciæc cost æ n è with the material of S n èwith the speciæc cost æ h è leads to the change in the total cost. we obtain æs = Væm + oèæmè lim S æ èm + æmè =S h + æs m! V =, ès n, S h è, + N rd èæè æ, : èè èè Variation of energy Let us compute the variation of energy in èè caused by the Weierstrass-type variation of properties æs. For simplicity, we consider such a variation that the axes of orthotropy of æs are codirected with the principal axes of the stress tensor èthe expression for æs is given in èèè. The increment of the energy æe is given by the quadratic form æeèæ æè =s T æsèæ æès è7è Degeneration In contrast to the two-dimensional case, in three dimensions we meet several degenerative cases. Let us discuss them. The variation of èè depends on two structural parameters æ and æ èor, in other notations, on æ and æè, such that æ æ æ + æ : If one of the æ i 's is zero or if æ + æ =, the structure degenerates into a second-rank laminate with parallel cylindrical inclusions. The eæective tensor S æ èmè of a second-rank laminate is similar to èè and is given by the formula: S æ èmè =S h + m, ès n, S h è, +è, mèn nd i èæè æ, Ni nd èæè is èè N nd i =èæ j, æ ij æ i èn j i j = : èè N j are deæned in è7è, and j is index of summation. Note that Einstein summation is used in èè, i.e., there is a summation on the repeated index j æ ij is the Dirak æ-function: if i = j æ ij = èè if i = j: If both æ and æ are zero èor if one of them is equal to oneè, the structure degenerates into a ærst-rank laminate for which Ni st èæè =N. Calculations of æs these degenerative variations are performed similarly to èè, using N nd computed in the expressions èè or N st, respectively. and the total cost of the variation is æj æm =èæ h, æ n + æeèæ æèè æm: è8è The increment è7è depends on the shape of the inclusions, speciæcally, on control parameters æ and æ èelongationsè of the inclusions in the laminates. In the degenerative cases, the number of the parameters decreases. Orientation of the inclusions As in two-dimensional case, we assume that the directions of laminates in the trial structure are codirected with the principal axes of the stress tensor. We codirect our labor coordinate system with the principle axes of the stress this yields to s = s = s =: Substituting the expressions for s and æsèæ æè èsee èè and èèè into è7è transforms the increment in the form: æeèæ æè = s T æsèæ æès = s T Vsæm = V s + V s + V s +V s s +V s s +V s s æm: è9è One can check that the coeæcients Vi depend on controls æ and æ as follows: Vi = K V i è h n h n èæ + V i è h n h n èææ +V i è h n h n èæ èè K is a quadratic polynomial of æ and æ.

5 . The Most Sensitive Variations The analysis is similar to the two-dimensional case. If the design is optimal, then all variations lead to the nonnegative increment æj : æj ès i S j j è=æ h, æ n + æeèæ æè 8æ æ æ èè æj ès i S j i èisavariation caused by adding material S j into material S i i is the æeld in material S i, and the set æ is æ = fæ æ j æé æé and æ + æ =g: èè If the condition èè is violated, then the cost is reduced by the variation, and the design fails the test. Therefore, the optimal èmost ëdangerous"è increment of energy è7è is E = min æææ æ Eèæ æè: If E é èè èè then the increment æj é for any variation and the structure satisæes the test... Calculations of Optimal Parameters Since the geometric shape of the variation is adjusted to the æeld, we consider optimal parameters æ and æ as functions of. Here we compute the derivative of æeèæ æè given in è7è, ænd optimal parameters, and calculate the optimal increment. First we compute the derivative of the increment with respect to æ and æ in the form: Eèæ æè = æè s s =,s æè Vèæ æè Vèæ æès T C, = V introduce a new variable v: Vèæ æès T = v èè èè v =èv v v v v v è. Since the variables s s and s are equal to zero, v v and v are zeros too, as it follows from èè. Therefore, we will not pursue the terms involving v v and v hereafter. After substituting the new variable v into èè, we obtain Eèæ æè =,v rd èæ æè v è7è which amounts to the following quadratic equation for v v rd èæ æè v =: è8è Now we can repeat the same calculations for the derivative ofæeèæ æè with respect to æ which amounts to the second quadratic equation: v rd èæ æè v =: è9è The next step is to solve the system of these equations è8è and è9è to determine the vector v. Substituting this solution into èè provides optimal values for æ and æ. We ænd that v belongs to one of the following sets V = V = vjv = v v = v èè and V = U =, vjv = U v = v v = v èè v h +v h, v h, v h : èè èv, v èè h, h è When one substitutes the vectors from V into èè, the optimal æ's and æ's are obtained as follows æ i = N æi æ i = N æi D i D i èè N æi N æi D i for i = are deæned in Appendix A, and æ and æ are from the set æ given in èè. Similarly, when vectors from set V are substituted into èè, the following optimal values of æ's and æ's are obtained æ j = N æj æ j = N æj D j D j èè N æj N æj D j for j = are deæned in Appendix A, and æ and æ are from the set æ given in èè. Degeneration If one of the parameters æ i or æ i computed in èè and èè is equal to zero or negative or if their sum is greater than one, then the optimal variation corresponds to a second-rank laminates. In this case, computation of the optimal variation æeèæè follows from

6 è7è èè is used for the calculation of èè. Consequently, taking the ærst derivative of æeèæè with respect to æ gives the following minimizing values of æ for different cases. Case A: If æ =, then æ = æ, æ =, æ in èè. Thus, the optimal æ is given by æ R = A x + B y + C z D è x + y è+c z æ R = B x è x, y è + C y è x, y è + A z è x, y è : èè Case B: If æ =, then æ = æ, æ =, æ in èè. Thus, the optimal æ is given by æ R = A x + C y + B z D è x + z è+c y æ R = B x è x, z è + A y è x, z è + C z è x, z è : èè Case C: If æ =, then æ = æ, æ =, æ in èè. Thus, the optimal æ is given by æ R = C x + A y + B z C x + D è y + z è æ R = A x è y, z è + B y è y, z è + A = h n + n h + n h C z è y, z è B = n h, n h, h h A = è h + n + h èè, n h + n h è h è, h + n èè n, h, h + n è B B = h è, n + h èè, h + n, h + n è C C = h è n, h èè n, h, h + n è B = è, h n +è n h, n n + h n è h, n h è, h + n + n èè C = èè n, h è h +è n n + h n, 8 h, n h è h, n h è n, h èè C = è n h, n h è D = è n h, h h + n h è: è7è è8è Here, æ Rj i denotes the optimal value of æ i when æ j = and the variation degenerates into a second-rank laminate. The energy increment is obtained by substitution of the optimal values æ Rj i into è7è. The optimal variation becomes a ærst-rank laminate if two of the optimal values of æ i are zero or negative.. Necessary Conditions The described condition èè is satisæed in permit a region of æelds in materials, and they are violated if the æelds are outside of this permitted region. To describe the permitted regions we use the results of the Section... The range of admissible æelds in an optimal structure is derived similarly to the two-dimensional case. In this section, we consider a well-ordered case, é and é.weæxthevolume fraction of the ærst material m = m in the following calculations.. Suppose that a trial inænitesimal inclusion of the second èstrongè material S is placed into the domain æ occupied with the ærst èweakè material. The necessary condition æj ès S è is obtained by formula èè as æj ès S è=æ, æ + EèS S è è9è is the tested æeld in the domain æ. To calculate EèS S è, we use the algorithm given in Appendix ècè. The inequality è9è depends only on the stress æeld since S and S are given. The set of æelds that satisæes the condition è9è is called F. If F, then the weak material S may be optimal in the sense that it cannot be denoted by the described variation. When the increment due to the inclusion of weak material into the strong material is calculated, the formula èè is used subscript h is.. Similarly we analyze the necessary condition of optimality of the èstrongè material S. The corresponding increment æj ès S è is due to the inserting of a trial inclusion of the ærst èweakè material S into the domain æ occupied by the second èstrongè material the condition is given in èè. æj ès S è=æ, æ + EèS S è èè is the æeld at a point of the domain æ.to calculate EèS S èwe use the algorithm given in Appendix ècè. The set of admissible æelds that satisæes the condition èè is called F.If F, then the strong material S is optimal in the sense that it cannot be denoted by the described variation. Remark When the increment due to the inclusion of strong material into the weak material is calculated, the formula èè becomes æs =,Væm+ oèæmè V =, ès, S è, + æ æ, subscript h in èè is equal to one.

7 7 8 8 σz Assembly of bounds: The interior region (intersection of ellipsoids) de nes the bound of the set F (stress in the weak material), the outer region de nes the zone F of optimality of the stress in the strong material, and the region between de nes the forbidden region. One of the materials has zero Poisson ratio. Fig. The union F = F [F of the two permitted sets does not coincide with the whole range of. The remaining part is called the forbidden region, Ff. In this region, none of the materials is optimal. An optimal structure should be constructed in such a way that the local elds never belong to Ff no matter what the external loading is. The calculation of the set F is based on the calculations of the sets F and F in each material S and S determined by (9) and (), respectively. σy The black triangular region is the third-rank laminate. White and gray regions show the second and rst-rank laminates are optimal, respectively. Both of the materials have nonzero Poisson ratio. σ z σy () () triplets i j and k corresponds to some directions of x y z axes. The constants are determined by the materials' elastic properties they correspond to optimal values of E( ) at =, or =. The analysis above shows that the boundary of F is an composition of three parts: An elliptical, cylindrical and plane part. The elliptical and the cylindrical parts respectively are given by the following expressions E σx. Range of Admissible Fields = A i + B ( j + k ) = = A i + B (j j j + j k j) = Fig. Results E σx The interior region of the small ellipsoid de nes the bound of the set F, and the region between the ellipsoids de nes the forbidden region when one of the materials has nonzero Poisson ratio. Fig. Remark The formula () shows the case when Poisson ratio of materials is zero. This formula is modi ed when the materials have an arbitrary Poisson ratio. In this case, the cylindrical and ellipsoidal parts are inclined on the angle de ned by the Poisson ration.

8 8 σz σz σy σy σx σx The black triangular region is the third-rank laminate. White and gray regions show the second and rst-rank laminates are optimal, respectively. Fig. The interior region of the small ellipsoid de nes the bound of the set F, and the region between the ellipsoids de nes the forbidden region when one of the materials is void. Fig. σz 8 σy σx 8 σy 8 σx 8 The intersection of three ellipsoids de nes the bound of the set F when one of the materials is void. Fig. The plane component of the boundary of F is: E = D (j x j + j y j + j z j) j = () the constant coe cient D is determined by the material constants and it corresponds to stationary values of and in (),(). Thus, the boundaries of the forbidden region has three components the plane, cylindrical and elliptical segments. Note that the components E = E = and E = together de ne the sur- Fig. 7 Two-dimensional case Region F lies inside of the region given by crosses, while region F lies outside of the region given by circles. The forbidden region Ff lies between the regions. face which are a rotationally invariant norm of the stress tensor. The graph of the permitted elds is presented in Figure {Figure for three di erent pairs of values for elastic moduli one of the two materials has zero Poisson ratio, both materials have nonzero Poisson ratio and one of the materials is void. Brie y, the weak material is present

9 9 at the intersection of the three parts mentioned above, and the strong material is observed at the union of three components. Now we can formulate the underlying principle of stiæness optimization by a two-material structure.. Optimal Structures A three-dimensional generalization of the ægure obtained for two dimensions in Figure 7 is given in Figure, and Figure. In Figure 7, we observed Cherkaev and Kíucíuk èè that region F lies inside of the region given by crosses, while region F lies outside of the region given by circles and the forbidden region Ff lies between the regions. Similarly, the region of F in three-dimensional case is observed as the inner region of the small ellipsoids in Figure, Figure and Figure and the region of F is observed as the outer part of the ellipsoids in Figure, and Figure. Consequently, the forbidden region Ff appears as a region between the big and small ellipsoids in the ægures of Figure, Figure and Figure. The optimal structures are not unique in contrast with the regions of optimal æelds. It is a known set of optimal geometries that provides the æelds inside of them on the boundary of the permitted regions due to adjustment of the inner parameters. These are the same third-rank laminates that we have described earlier. Suppose that an optimal structure is submerged into a homogeneous external æeld. If the external æeld belongs to F or F, then the optimal design consists of one material S or S. The æeld jumps over the forbidden region along the boundary surface between zones occupied with the strong and weak materials. It turns out that such jumps are possible only if the boundary becomes a curve with inænitely many wiggles, see Cherkaev èè. As a result, the optimal structure becomes a composite in which the æelds belong to the boundaries of F = constant or F = constant at each point. To provide this feature, the volume fraction and the inner parameters vary together with the average stress æeld. Fields in optimal third-rank laminates If the external æeld belongs to the pyramid supported by the black triangular regions in the Figure, Figure, and Figure 8 then the optimal structure corresponds to nondegenerative laminates of the third rank. The directions of laminates are determined by the eigenvectors of the stress. The anisotropy of the structure balances resistance against stresses acting in the orthogonal directions with diæerent magnitudes. The æelds in the layers of the ærst, second, and third rank in the strong èwrappingè materials correspond to three points on the plane triangles shown in black, the æeld in the ærst èinnerè layer corresponds to the vertex of the triangle, the æeld in the Fig. 8 One of the materials is void. The boundary of the permitted region of the strong material. Observe that the components of the boundary that correspond to optimality of simple laminates disappear. second layer corresponds to a point on a side of it, and the æeld in the outer layer corresponds to the point inside the triangle. The corresponding æeld in the nucleus belongs to the corner of the permitted region èintersection of the three ellipsoidsè: It satisæes the condition j x j = j y j = j z j = Constant: The corresponding region is shown on Figure, Figure, and Figure. Fields in optimal second-rank laminates When one of the eigenvalues of the stress tensor is signiæcantly larger than the others, see èè,è8è, the optimal structure degenerates into a second-rank laminate. These regions are shown in white in Figure, and Figure. The generator of optimal cylindrical inclusions is codirected with maximal eigenstress. Other eigenvectors corresponding to two smaller eigenvalues that determine the normals to the layers in the second-rank laminate. The maximal possible stiæness of this anisotropic structure is codirected with the generator the optimal structure adjusts itself to equalize the response to stresses of diæerent magnitudes applied in the directions across the generator. The stresses in the wrapping layer correspond to two points along the generator, one of them on the boundary of this domain. they satisfy the relation j x j + j y j = constantèxè é j z j subindex z shows the direction of the generator of cylinders. The corresponding stress in the weak material

10 we show the results for the zero Poisson ratio material to keep the notations simple. Deæne the norm kk T of the stress æeld as following: kk T = jx j + j y j + j z j if j x j + j y jj z j èj x j + j y jè + j z j if j x j + j y j é j z j: èè In an optimally designed body, the following conditions are hold inside of the material: kk T = 8 é : éc if the material is solid, = C if the material is involved in an optimal structure. èè Fig. 9 Permitted regions in badly-ordered case. inside the cylinder belongs to the rib of its domain F It satisæes the condition j x j = j y j é j z j: Fields in optimal simple laminates The æelds in optimal laminate belong to the points of elliptical surface. The æelds in both materials are constant, and they correspond to regular points on the elliptical components of the boundaries of the permitted sets. Badly-ordered case When available materials are badlyordered, the admissible set of stresses is restricted by elliptic hyperboloids, see Figure 9. The hyperbolic segments of the boundaries replace spherical and cylindrical segments of the boundaries for well-ordered case. In this case, the preferable material is deæned not by the intensity of the loading, but by its type: an intensive shear loading requires the material with larger shear modulus, and an intensive bulk loading requires the other material with larger bulk modulus.. Topology Optimization The asymptotic case when one of the materials is void is of special interest. The optimal design problem becomes a problem of determining of the shape of the material frame in the design domain, or determining the number and location of holes èvoidsè in a solid structure. This problem is often called topology optimization problem. Our results can be easily reformulated for this case. Here C is a positive constraint that depends on the amount of given material and of the intensity of external loading, èsee Figure 8è. The formulated result realizes the centuries-old rule of rational design: the material should never be understressed. The material, however, may be overstressed since we cannot do better that place the solid material in the intensively stressed domain. The optimal structures that realize this requirement are again non-unique, see for discussion in Cherkaev èè. In particular, the laminate can be used as optimal structures. The ærst-rank èsimpleè laminate is never optimal since the structure would break apart in this asymptotical case. The properly adjusted second-rank laminates correspond to cylindrical parts of the surface in Figure 8. The third-rank laminate correspond to plane triangular regions. The volume fraction, orientation, and the geometric parameters of structures vary to make the æelds in the material satisfy the conditions èè. A Terms Used in Section :: In this appendix, we give explicit formulas for the terms and an algorithm used in Section... The terms used in èè and èè is given as B = è,v +v v, v, v v +v v è h, èv v +v v, v, v v, v è h h èè,8, v v, èv v + v v è, èv + v è æ h h: N æ = è h h + n h + n h è x, è z + y èè h h, n h + n h è N æ = è h h + n h + n h è y, è x + z èè h h, n h + n h è è7è è8è

11 D = h è, h + n èè y + x + z è è9è N æ =,è h, n èè h + h èè h + n + h è x + èè n, h è h +è, h n, 8 h + n n + n h è h + h n è, n + h èè y + N æ = èè, n +8 h è h, è n h +è n, h è n è h, n h è, h + n èè x, èè n, h è h + è n n, n h + n h, 8 hè h, n h è, h + n èèè y + z è èè N æ èè n, 8 h è h + è n h, h, 9 n n, h n è h +è h + n è n h è z èè = èè8 h, n è h +è n h, n h, n n è h, n h è n, h èè y + èè8 h, n è h +è, n h + n h + h + 9 n n è h, n h è h + n èèè x + z è è7è N æ = èè n, 8 h è h, è n h, n h +9 n n + hè h + n h è h + n èè x +è h + h è è h + h + n èè n, h è y +è8 h + è n, n + h è h, è9 n n +9 n h + n h, hè h + n h è h + n èè z èè D = è h, n èè h, n èèè h + h èè x + z è B Formulas for Energies,è h, h è y è: è8è D = è, n + h èè, h + n èèè h, h è x,è y + z èè h + h èè: èè The energy E R i for the ærst-rank laminate æ i = is deæned as follows E R = A x + Bè y + zè, Cè y + z è x + D y z 8 h è n + n è h N æ =,è, n + h èè h + h èè h + n + h è x + è8 h, è8 n, n, h è h +è h, 9 n h, h n, 9 n n è h è n + h è n h è y + èè n, 8 h è h +è,9 n n + n h, h n, hè h + è n + h è n h è z èè E R = Bè x + zè+a y, Cè x + z è y + D x z 8 h è n + n è h E R = Bè x + yè+a z, Cè x + y è z + D x y 8 h è n + n è h N æ = èè h + n è h +è,9 n h + n n, n h + hè h + h è, n n + n h, n h + hèè x, è h, n èè h + h èè h + h + n è y + èè n, 8 h è h +è n h, h, 9 n n, h n è h +è h + n è n h è z èè A = è9è h, n è n, h n +7 hè h, h è n h, h n, n n è h, n h n B = è,9è n + n n, n h è+ h n è h + h è h n + n h, n n è h, n h n C = è h + h èè h n h, h h n, h n n + n h n è D = è8 n, h n +9è n, h è n è h + h èè h, n è n + h n è h, 8 n h n D = è h, n èè h, n èèè h + h èè x + y è,è h, h è z è èè The energy E R i for the third-rank laminate all æ i 's in èè are in the interval of è è is deæned as follows

12 Elseif æ = then æ = æ R and æ = æ R = è h + h èè y + x + z è è h, n è 8 h è n + h è E R E R = AA DD E R = AA DD E R = AA DD èè h + h èè z + y è+è h, h è x è 8 h h èè, h + h è z +è h + h èè x + y èè 8 h h èè h + h èè x + z è+è h, h è y è 8 h h If æ è è then E R Elseif æ è è then E R Elseif æ =and æ = then E R Elseif æ =and æ = then E R DD = è, h + è n, h, n è h +7 n n + n h, h + n h AA = è h + h èè n, h èè n, h è: end Elseif æ = æ =and æ = then E R C Algorithm to Find Sets of Fields In this section, we give analgorithm to obtain sets F given in è9è and F given in èè for each material. Brieæy, the optimal energy is calculated through the following algorithm for any given material properties, not necessarily restricted to well-ordered case. E = procè i S j S i è If æ æ and æ è è E R Elseif æ = then æ = æ R and æ = æ R Elseif æ = then æ = æ R and æ = æ R If æ è è then E R end Elseif æ è è then E R Elseif æ =and æ = then E R Elseif æ =and æ = then E R If æ è è then E R Elseif æ = end æ =and æ = then E R end Elseif æ = Elseif æ è è then E R Elseif æ =and æ = then E R Elseif æ =and æ = then E R æ =and æ = then E R The energies used in this algorithm and their notational explanations are given in è9è, è9è. References Allaire, G., Bonnetier, E., Francfort, G. A., and Jouve, F.: 997, Numer. Math. 7èè, 7 Cherkaev, A. V.:, Variational Methods for Structural Optimization, Vol., Springer-Verlag, New York Berlin Heidelberg Cherkaev, A. V. and Kíucíuk, I.:, Structural Optimization, Submitted

13 Cherkaev, A. V. and Palais, R.: 997, Structural Optimization èè, Francfort, G. and Murat, F.: 98, Arch. Rat. Mech. Anal. 9, 7 Gibiansky, L. V. and Cherkaev, A. V.: 98, Design of Composite Plates of Extremal Rigidity, Report 9, Ioæe Physicotechnical Institute, Acad of Sc, USSR, Leningrad, USSR, English translation in Gibiansky and Cherkaev è997aè Gibiansky, L. V. and Cherkaev, A. V.: 987, Microstructures of Composites of Extremal Rigidity and Exact Estimates of the Associated Energy Density, Report, Ioæe Physico- Technical Institute, Acad of Sc, USSR, Leningrad, USSR, English translation in Gibiansky and Cherkaev è997bè Gibiansky, L. V. and Cherkaev, A. V.: 997a, in A. V. Cherkaev and R. V. Kohn èeds.è, Topics in the Mathematical Modelling of Composite Materials, Vol. of Progress in nonlinear diæerential equations and their applications, pp 9í7, Birkhíauser Boston, Boston, MA, Translation. The original publication in Gibiansky and Cherkaev è98è Gibiansky, L. V. and Cherkaev, A. V.: 997b, in A. V. Cherkaev and R. V. Kohn èeds.è, Topics in the Mathematical Modelling of Composite Materials, Vol. of Progress in nonlinear diæerential equations and their applications, pp 7í7, Birkhíauser Boston, Boston, MA Kíucíuk, I.:, Ph.D. thesis, University ofutah Lipton, R. and Diaz, A.: 99, in N. Olhoæ and G. Rozvany èeds.è, Structural and Multidisciplinary Optimization, pp í8, Pergamon, Oxford, UK, 99, Goslar, Germany Lurie, K. A.: 97, Optimal Control in Problems of Mathematical Physics, Nauka, Moscow, Russia, in Russian Lurie, K. A.: 99, Applied Optimal Control Theory of Distributed Systems, Plenum Press, New York,USA, New York Olhoæ, N., Scheel, J., and Rnholt, E.: 998, Structural Optimization,