Sławomir Czarnecki - Academia.edu (original) (raw)

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Papers by Sławomir Czarnecki

Advances in Structural and Multidisciplinary Optimization, 2017

The paper deals with the free material design and its constrained versions constructed by imposin... more The paper deals with the free material design and its constrained versions constructed by imposing: (a) cubic symmetry (cubic material design, CMD), (b) isotropy with: (b1) independent bulk and shear moduli (isotropic material design, IMD), and (b2) fixed Poisson’s ratio (Young’s modulus design, YMD). In the latter case the Young modulus is the only design variable. The moduli are viewed as non-negative, thus allowing for the appearance of void domains within the design domain. The paper shows that all these methods (CMD, IMD, YMD) reduce to two mutually dual problems:

physica status solidi (b), 2020

The article discusses the 2D problem of manufacturability of the minimum compliance designs of th... more The article discusses the 2D problem of manufacturability of the minimum compliance designs of the structural elements made of inhomogeneous materials of local isotropic or square symmetry properties. The available isotropic material design (IMD) and cubic material design (CMD) methods deliver the optimal distribution of the elastic moduli within the design domain. Within the 2D setting, the cubic symmetry reduces to the symmetry of a square. The varying underlying microstructures corresponding to the optimal designs are recovered by matching the values of the optimal moduli with the values of the effective moduli of the representative volume elements (RVEs) computed by the asymptotic homogenization method for periodic media. The shape of the RVE and its internal symmetries are properly selected providing assumed isotropy or symmetry of a square of the homogenized constitutive tensor. The microstructure topology is described by parametric description of single (or several) fibers in...

Mechanics Research Communications, 2018

Materials, 2017

The paper discusses the problem of manufacturability of the minimum compliance designs of the str... more The paper discusses the problem of manufacturability of the minimum compliance designs of the structural elements made of two kinds of inhomogeneous materials: the isotropic and cubic. In both the cases the unit cost of the design is assumed as equal to the trace of the Hooke tensor. The Isotropic Material Design (IMD) delivers the optimal distribution of the bulk and shear moduli within the design domain. The Cubic Material Design (CMD) leads to the optimal material orientation and optimal distribution of the invariant moduli in the body made of the material of cubic symmetry. The present paper proves that the varying underlying microstructures (i.e., the representative volume elements (RVE) constructed of one or two isotropic materials) corresponding to the optimal designs constructed by IMD and CMD methods can be recovered by matching the values of the optimal moduli with the values of the effective moduli of the RVE computed by the theory of homogenization. The CMD method leads to a larger set of results, i.e., the set of pairs of optimal moduli. Moreover, special attention is focused on proper recovery of the microstructures in the auxetic sub-domains of the optimal designs.

physica status solidi (b), 2017

The stiffest structural elements to resist given systems of loads of prescribed weights are optim... more The stiffest structural elements to resist given systems of loads of prescribed weights are optimally formed from an inhomogeneous isotropic material. The products are cut out from the design domain, thus solving the shape design problem simultaneously. The optimal designs reflect the weights of the loads, see the illustrative design for mutually orthogonal loads of different weights.

Mechanics Research Communications, 2016

The influence of the loading conditions on the trabecular architecture of a femur is investigated... more The influence of the loading conditions on the trabecular architecture of a femur is investigated by using topology optimization methods. The response of the bone to physiological loads results in changes of the internal architecture of bone, reflected by a modification of internal effective density and mechanical properties. The homogenization based optimization model is developed for predicting optimal bone density distribution, wherein bone tissue is assumed to be a composite material consisting of a mixture of material and void. The homogenization scheme treats the geometric parameters of the microstructures and their orientation as design variables and homogenizes the properties in that microstructure, which is generally anisotropic. The penalization of the optimal material density then leads to a classical optimal structure which consists of regions with bone material and regions without bone material. The IMD (Isotropic Material Design) approach is next applied to determine the optimal elasticity tensor in terms of the bulk and shear moduli as well for the present loading applied to the femoral bone sample. IMD is able to provide both the external shape and topology together with the optimal layout of the isotropic moduli. Both topology optimization methods appear to be complementary. Simulations of the internal bone architecture of the human proximal femur results in a density distribution pattern with good consistency with that of the real bone.

physica status solidi (b), 2015

Structural and Multidisciplinary Optimization, 2014

The paper deals with minimization of the weighted sum of compliances related to the load cases ap... more The paper deals with minimization of the weighted sum of compliances related to the load cases applied non-simultaneously. The design variables are all components of the Hooke tensor, subject to the isoperimetric condition bounding the integral of the sum of the Kelvin moduli. This free material design problem is reduced to an equilibrium problem-in two formulations-of an effective body with locking. The stress-based formulation reduces to minimization of an integral of a certain norm of stress fields over the stress fields which equilibrate the given loads. The equivalent displacement-based formulation involves a locking locus defined by using a norm being dual to the previous one. The optimal Hooke tensor is determined by using the stress fields solving the auxiliary locking problem. To make the optimal Hooke tensor positive definite one should consider at least 3 load conditions in the 2D case and not less than 6 load conditions in the 3D case.

Structural and Multidisciplinary Optimization, 2013

The paper deals with two minimum compliance problems of variable thickness plates subject to an i... more The paper deals with two minimum compliance problems of variable thickness plates subject to an in-plane loading or to a transverse loading. The first of this problem (called also the variable thickness sheet problem) is reduced to the locking material problem in its stress-based setting, thus interrelating the stress-based formulation by Allaire (2002) with the kinematic formulation of Golay and Seppecher (Eur J Mech A Solids 20:631-644, 2001). The second problem concerning the Kirchhoff plates of varying thickness is reduced to a non-convex problem in which the integrand of the minimized functional is the square root of the norm of the density energy expressed in terms of the bending moments. This proves that the problem cannot be interpreted as a problem of equilibrium of a locking material. Both formulations discussed need the numerical treatment in which stresses (bending moments) are the main unknowns.

Computer Methods in Applied Mechanics and Engineering, 2011

The paper deals with optimal design of linearly elastic plates of the Kelvin moduli being distrib... more The paper deals with optimal design of linearly elastic plates of the Kelvin moduli being distributed according to a given pattern. The case of two loading conditions is discussed. The optimal plate is characterized by the minimum value of the weighted sum of the compliances corresponding to the two kinds of loads. The problem is reduced to the equilibrium problem of a hyperelastic mixture of properties expressed in terms of two stress fields. The stress-based formulation (P) is rearranged to the displacement-based form (P∗). The latter formulation turns out to be well-posed due to convexity of the relevant potential expressed in terms of strains. Due to monotonicity of the stress–strain relations the problem (P∗) is tractable by the finite element method, using special Newton’s solvers. Exemplary numerical results are presented delivering layouts of variation of elastic characteristics for selected values of the weighting factors corresponding to two kinds of loadings.

Computer Methods in Applied Mechanics and Engineering, 2008

ABSTRACT The subject of the paper is an optimal choice of material parameters characterizing the ... more ABSTRACT The subject of the paper is an optimal choice of material parameters characterizing the core layer of sandwich plates within the framework of the conventional plate theory in which the core layer is treated as soft in the in-plane direction. The mathematical description is similar to the Hencky–Reissner model of plates with transverse shear deformation. Here, however, the bending stiffnesses and the transverse shear stiffnesses can be designed independently. The present paper deals only with optimal design of the core layer to make the plate compliance minimal. Two core materials are at our disposal, which leads to the ill-posed problem. To consider it one should relax this problem by admitting composite domains and characterize their overall properties by the homogenization formulae. The numerical approach is based on this relaxed formulation thus making it mesh-independent. The equilibrium problem is solved by the DSG3 finite element method. The optimization results are found with using the convergent updating schemes of the COC method.

Bulletin of the Polish Academy of Sciences: Technical Sciences, 2013

Optimization of structural topology, called briefly: topology optimization, is a relatively new b... more Optimization of structural topology, called briefly: topology optimization, is a relatively new branch of structural optimization. Its aim is to create optimal structures, instead of correcting the dimensions or changing the shapes of initial designs. For being able to create the structure, one should have a possibility to handle the members of zero stiffness or admit the material of singular constitutive properties, i.e. void. In the present paper, four fundamental problems of topology optimization are discussed: Michell’s structures, two-material layout problem in light of the relaxation by homogenization theory, optimal shape design and the free material design. Their features are disclosed by presenting results for selected problems concerning the same feasible domain, boundary conditions and applied loading. This discussion provides a short introduction into current topics of topology optimization

physica status solidi (b), 2018

The paper discusses the problem of recovery of the microstructure of the least compliant bodies o... more The paper discusses the problem of recovery of the microstructure of the least compliant bodies of non‐homogeneous optimal isotropic properties predicted by the Isotropic Material Design method. The three‐dimensional microstructure is assumed as constructed by a subsequent lamination process in which two isotropic materials of ordered properties are used, the process being repeated three or six times, subsequently. The Hooke tensors closest to the optimal ones are found by making use of the analytical Francfort–Murat formulae for effective moduli of third and sixth rank sequential laminates. The ability to model an auxetic behavior within the subdomains where the optimal Poisson's ratio assumes negative values is also shown.