Institute of Fundamental Technological Research

In previous works, the authors introduced a multiscale optimization method to maximize the stiffness of elastic solids with biomimetic cancellous microstructures described by a finite set of parameters. Although effective, the procedure is computationally expensive when solving large-scale problems using per-element non-linear constraints to impose local bounds on the solid volume fraction. This work improves the computational performance of the method by exploring two strategies to completely dispense with nonlinear local constraints: to bound the microparameters so the microsctructures are always within the solid fraction of trabecular bone, and to map the microparameters onto an auxiliary set of parameters that are linearly bounded. As a side effect, the design spaces are reduced. Such reductions are assessed in terms of the bulk and shear moduli and elastic symmetries, which are compared to those of natural bone. Performances of the two strategies are assessed by solving a series of benchmark problems and studying the stiffness of a hip prosthesis. The strategy based on the isoparametric mapping achieves the best results, performing up to 2000 times faster while marginally reducing the design space. Thus, the isoparametric mapping approach makes the multiscale design method a suitable tool for solving large-scale problems of practical interest.

Multiscale optimization , Trabecular bone , Parameterized microstructures , Computational performance , Large-scale problems

Cancellous bone is a highly porous, heterogeneous, and anisotropic material which can be found at the epiphyses of long bones and in the vertebral bodies. The hierarchical architecture makes cancellous bone a prime example of a lightweight natural material that combines strength with toughness. Better understanding the mechanics of cancellous bone is of interest for the diagnosis of bone diseases, the evaluation of the risk of fracture, and for the design of artificial bones and bone scaffolds for tissue engineering. A multiscale optimization method to maximize the stiffness of artificial bones using biomimetic cellular microstructures described by a finite set of geometrical micro-parameters is presented here. The most outstanding characteristics of its implementation are the use of: an interior point optimization algorithm, a precalculated response surface methodology for the evaluation of the elastic tensor of the microstructure as an analytical function of the micro-parameters, and the adjoint method for the computation of the sensitivity of the macroscopic mechanical response to the variation of the micro-parameters. The performance and effectiveness of the tool are evaluated by solving a problem that consists in finding the optimal distribution of the microstructures for a proximal end of a femur subjected to physiological loads. Two strategies for the specification of the solid volume fraction constraints are assessed. The results are compared with data of a computed tomography study of an actual human bone. The model successfully predicts the main features of the spatial arrangement of the trabecular and cortical microstructures of the natural bone.

multiscale optimization, cancellous bone, bone scaffolds, parameterized microstructures

This report presents the modeling of pressure-assisted sintering within the framework of a multiscale approach. Three individual numerical methods have been collectively applied to predict the behavior of a sintering body at three different scales. The appropriate solutions to connect each model/scale have been proposed. Molecular dynamics have been employed to evaluate the grain boundary diffusion coefficient at the atomistic scale. The obtained results of diffusive parameters have been transferred to the micromechanical model of sintering. Here, the discrete element method was used to represent the sintered material properties at the microscopic scale. Micromechanical based results have been validated by own experimental data of material density evolution, indicating the required coincidence. The transfer from micro- to the macroscopic model has been realized by determining the macroscopic viscous moduli from discrete element simulations and subsequently applying them to the continuum model of sintering. The numerical results from finite element simulations at the macroscopic scale have been compared with discrete element ones.

sintering, multiscale modeling, discrete element method, molecular dynamics, finite element method

Natural (or biological) materials usually achieve outstanding mechanical performances. In particular, cancellous bone presents a high stiffness/strength toweight ratio, so its structure inspires the development of novel ultra-light cellularmaterials. Amultiscale method for the design of elastic solids with a cancellous bone parameterized biomimetic microstructure is introduced in this work. The method combines a finite element model to evaluate the stiffness of the body at the macroscale with a gradient-based nonlinear constrained optimization solver to obtain the optimal values of themicroparameters andmicrostructure orientation over the body domain. The most salient features of the implementation are an offline response surface methodology for the evaluation of the microstructure elastic tensor in terms of the microparameters, an adjoint method for the computation of the sensitivity of the macroscopic stiffness to the microparameters, a quasi-Newton approximation for the evaluation of the Hessian matrix of the nonlinear optimizer, and a distanceweighted filter of the microparameters to remediate checkerboard effects. The settings of the above features, the optimizer termination options, and the initial values of the microparameters are investigated for the best performance of the method. The effectiveness of the method is demonstrated for several examples, whose results are compared with the reference solutions calculated using a SIMP method. The method shows to be effective; it attains results coherent with SIMP approaches in terms of stiffness and spatial material distribution. The good performance of themultiscalemethod is attributed to the capability of the parameterized mimeticmicrostructure to attain bulk and shear moduli that are close to the Hashin-Shtrikman upper bounds over the complete solid volume fraction range.

multiscale optimization, cancellous bone, parameterized microstructure, interior-point optimizer, biomimetic materials

Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (Comput Methods Biomech Biomed Eng 9:135–147, 2006. doi:10.1080/10255840600751473) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples.

Cancellous bone, Parameterized microstructure, Elastic properties, Homogenization, Symmetry classes, Optimization

The Fast Fourier Transform–based method, originally introduced by Moulinec and Suquet in 1994 has gained popularity for computing homogenized properties of composites. In this work, the method is used for the computational homogenization of the elastic properties of cancellous bone. To the authors' knowledge, this is the first study where the Fast Fourier Transform scheme is applied to bone mechanics. The performance of the method is analyzed for artificial and natural bone samples of 2 species: bovine femoral heads and implanted femurs of Hokkaido rats. Model geometries are constructed using data from X‐ray tomographies, and the bone tissue elastic properties are measured using microindentation and nanoindentation tests. Computed results are in excellent agreement with those available in the literature. The study shows the suitability of the method to accurately estimate the fully anisotropic elastic response of cancellous bone. Guidelines are provided for the construction of the models and the setting of the algorithm.

accelerated FFT method, cancellous bone, homogenized elastic properties

A computational model that allows to explicitly determine orthotropic elastic constants of plain-weave fabric-reinforced composite ply as functions of microstructure parameters has been developed in this study. These relationships are not given in the form of analytical formulae (as it is in the case of approximate analytical models) but in the form of an extensive database of numerically evaluated results for different microstructure instances and a numerical scheme that interpolates the results. To build the database, a standard finite-element-based homogenization technique of a periodic representative volume element is employed. As a result, a numerical algorithm is provided that may be easily employed in FE codes as a part of a regular constitutive subroutine. Sensitivity of the composite elastic constants with respect to the microstructure parameters is also directly available from the model.

fabric-reinforced composite, plain-weave fabric, homogenization, parametric constitutive equations, finite element analysis

Although there have been several numerical models used by different authors to model microstructural behaviour of a woven fabric-reinforced composite ply, they all suffer from geometric simplifications that may affect their validity or at least accuracy of results. This paper presents an enhanced model of a representative volume element RVE of this kind of composite with the plain-weave fabric pattern. Many important simplifications are overcome while the mathematical description remains simple. A dedicated FE mesh generator allows to reproduce all geometric details in the computational model. Computational examples of homogenization of orthotropic elastic constants illustrate application of the model.

geometric modelling, fabric-reinforced composite, plain-weave fabric, homogenization, finite element analysis

The NUMPRESS System has been developed in IPPT PAN as a result of a project financially supported by European Regional Development Fund (within the Innovative Economy Programme) and is dedicated to small and middle enterprises dealing with sheet metal forming. The program consists of (i) an analytical module for analysis of forming processes with the finite element method, (ii) an optimization module controlling execution of the analytical module and performing optimization with respect to selected process parameters, in both deterministic and robust formulation, (iii) a reliability analysis module controlling execution of the analytical module to assess how random distribution of design parameters affects forming results, and (iv) a graphical user interface enabling communication between modules and easy definition of design parameters and optimization criteria. The analytical module consists of two independent programs up to the user's choice: NUMPRESS-Flow, a faster and less accurate program for implicit quasi-static analysis of rigid-viscoplastic shells (based on the flow approach) and NUMPRESS-Explicit, a program for explicit dynamical analysis of elastic-plastic and elastic-viscoplastic shells. Both programs are interfaced to a well-known commercial graphical pre- and postprocessor GiD. Fundamentals of formulations employed in the system and numerical examples are presented in the paper.

sheet metal forming, finite element method, deterministic and robust design optimization, reliability analysis

The main objective of this paper is to improve stability conditions, uniqueness and convergence of numerical analysis of metal forming processes with contact constraints enforced by the penalty method. A commonly known drawback of this approach is the choice of penalty factor values. When assumed too low, they result in inaccurate fulfillment of the constraints while when assumed too high, they lead to ill-conditioning of the equations system which affects stability and uniqueness of the solution. The proposed modification of the penalty algorithm consists in adaptive estimation of the penalty factor values for the particular system of finite element equations and for the assumed allowed inaccuracy in fulfillment of the contact constraints. The algorithm is tested on realistic examples of sheet metal forming. The finite element code based on flow approach formulation (for rigid-plastic and rigid-viscoplastic material model) has been used.

Contact modeling, Penalty approach, Metal forming, Deep drawing

A computational model that allows to explicitly determine transversely isotropic elastic constants of uni-directional fiber–matrix composite tow as functions of microstructure parameters has been developed in this study. These relationships are not given in the form of analytical formulae (as it is in the case of approximate analytical models) but in the form of an extensive database of numerically evaluated results for different microstructure instances and a numerical scheme that interpolates the results. To build the database, a standard finite-element-based homogenization technique of a periodic RVE is employed. The technique is enhanced by introduction of averaging procedure over different shapes of the 2D fiber layout pattern in the tow cross-section. As a result, a numerical algorithm is provided that may be easily employed in FE codes as a part of a regular constitutive subroutine. Sensitivity of the composite elastic constants with respect to the microstructure parameters is also directly available from the model.

Uni-directional fiber–matrix composite, Parametric constitutive model, Finite element analysis, Homogenization, Representative volume element, Parameter sensitivity

A computational model of adaptive bone remodelling is formulated as an optimization problem of instantaneous changes in microstructure that minimize a functional describing the structure quality rate. Microstructure is locally described by a set of scalar geometric parameters. Macroscopic (continuum) elastic properties are assumed orthotropic and expressed as known functions of the geometric parameters. Strain energy is considered the quality measure of bone at given load conditions.
The instantaneous rate of geometric parameters is postulated to minimize the rate of the quality functional. An optimization problem is formulated in the continuum description and then it is discretized both in space and time. Numerical simulations predict bone remodelling for femur without and with a hip endoprosthesis.

Adaptive bone remodelling, Orthotropy, Finite element method, Optimization

The paper presents an analytical constitutive design sensitivity analysis (DSA) algorithm for explicit dynamics of elastic-plastic finite rotation shells. Two explicit dynamical algorithms for finite rotation shells are presented, and the DSA is developed for the one formulated in terms of the rotation vector and its time derivatives, {ψ,ψ˙,ψ¨}. The hypo-elastic constitutive model based on the Green-McInnis-Naghdi stress rate is used to derive an incremental algorithm in terms of ‘back-rotated’ objects. The associative deviatoric Huber-Mises plasticity modified by plane stress conditions is implemented in the form suitable for finite rotation/small elastic strain increments. The analytical DSA is developed for the above-specified problem, with the design derivatives calculated w.r.t. material parameters. Design-differentiation of the dynamic algorithm and the scheme of handling the history data and the predicted values in differentiation, which is crucial in computing correct derivatives, are described. Besides, we show how to avoid Newton loops in the DSA algorithm, when such a loop is present in the constitutive algorithm. Numerical examples show that, despite a great complexity of the solution algorithm for the finite-rotation elastic-plastic shells, it is feasible to compute analytical design derivatives of very good accuracy.

Explicit dynamics, Finite rotation shell, Elastic-plastic material, Analytical Design Sensitivity Analysis for constitutive parameters

Implicit time integration algorithm derived by Simo for his large-deformation elasto-plastic constitutive model is generalized, for the case of isotropy and associative flow rule, towards viscoplastic material behaviour and consistently differentiated with respect to its input parameters. Combining it with the general formulation of design sensitivity analysis (DSA) for non-linear finite element transient equilibrium problem, we come at a numerically efficient, closed-form finite element formulation of DSA for large deformation elasto-plastic and elasto-viscoplastic problems, with various types of design variables (material constants, shape parameters). The paper handles several specific issues, like the use of a non-algorithmic coefficient matrix or sensitivity discontinuities at points of instantaneous structural stiffness change. Computational examples demonstrate abilities of the formulation and quality of results.

design sensitivity analysis, finite element method, large deformations, plasticity, viscoplasticity

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of “equivalent” trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.

Cancellous bone, Equivalent microstructure, Constitutive modelling, Elastic properties, Orthotropy, Fabric

This paper concerns design sensitivity analysis (DSA) for an elasto–plastic material, with material parameters depending on, or serving as, design variables. The considered constitutive model is Huber–Mises deviatoric plasticity with non‐linear isotropic/kinematic hardening, one which is applicable to metals.

The standard radial return algorithm for linear hardening is generalized to account for non‐linear hardening functions. Two generalizations are presented; in both the non‐linearity is treated iteratively, but the iteration loop contains either a scalar equation or a group of tensorial equations. It is proven that the second formulation, which is the one used in some parallel codes, can be equivalently brought to a scalar form, more suitable for design differentiation. The design derivatives of both the algorithms are given explicitly, enabling thus calculation of the ‘explicit’ design derivative of stresses entering the global sensitivity equation.

The paper addresses several issues related to the implementation and testing of the DSA module; among them the concept of verification tests, both outside and inside a FE code, as well as the data handling implied by the algorithm. The numerical tests, which are used for verification of the DSA module, are described. They shed light on (a) the accuracy of the design derivatives, by comparison with finite difference computations and (b) the effect of the finite element formulation on the design derivatives for an isochoric plastic flow.

design sensitivity analysis, elasto–plastic material with non‐linear hardening, parallel finite element code

Evaluation of constitutive properties of cancellous bone and their relationships to microstructural parameters is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of ‘equivalent’ trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. Two cell types are analysed: cube- and prism-based. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with three shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Six static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Multi-point boundary conditions imposed on the models ensure mutual fitting of deformed neighbouring cells. Values of computed stresses allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement, though evidently better in the case of the prism-based cell model.

Cancellous bone, Equivalent microstructure, Constitutive modelling, Elastic properties, Orthotropy

Implant separation from bone tissue, resulting in the necessity for revision surgery, is a serious drawback of cementless total joint replacement. Unnatural stress distribution around the implant is considered the main reason for the failure. Optimization of the implant properties, especially its geometric parameters, is believed to be the right way to improve reliability of joint prosthetics. An efficient numerical model of the femur–implant system is presented in the paper, including the finite element formulation featuring computation of sensitivity gradients, parametric mesh generator, and a gradient-based optimization scheme. Numerical examples show results of shape optimization of an implant for two sets of design parameters and for the initial stability criterion taken as the optimization goal. The optimum shape appears to be relatively long and proximally porous-coated on about half of its length. The method can be flexibly adjusted to various implant types, stress- and displacement-based optimum criteria, and geometric design parameters.

finite element analysis, optimisation, prosthetics, bone, biomechanics

An algorithm for design sensitivity analysis of elasto-plastic response for the case of design-dependent geometry is discussed. The domain parametrization approach is employed. Some important implications for the concept of isoparametric element formulation are indicated. The idea of shape sensitivity computations is similar to that of sensitivity with respect to material parameters. However, derivation of additional geometric matrices and integral expressions is necessary. A possible discontinuity of sensitivity solutions is indicated and it is shown that the algorithm handles it correctly. A plane stress computational example illustrates the application of the method.

Development of computational models for plane stress elasto-plasticity is known to make some difficulties non-existent in 3D and plane strain formulations. Tedious derivation of plane stress algorithmic tangent matrices which cannot be directly obtained as a special case of 3D matrices may serve here as an example. In this paper specific features of plane stress computations are discussed in the context of parameter sensitivity analysis. Elasto-plastic material model with isotropic/kinematic hardening as well as two viscoplastic models are considered. Explicit expressions for sensitivity gradients are derived. Instructive examples solved by FEM illustrate the paper.

The paper presents equations and algorithms for numerical computation of elasto-plastic and elasto-viscoplastic constitutive parameter sensitivity problems. The general integration idea is based on the return-mapping algorithm. Two viscoplastic constitutive models are discussed in details: the overstress (Perzyna) model and the power law strain and strain-rate hardening model. The use of the consistent tangent operator is shown to be essential for the accuracy of the sensitivity analysis. A possible discontinuity of sensitivity at the transition (yield limit) point is discussed. It is concluded that in principle the nonuniqueness of the sensitivity solutions at such points does not invalidate the general idea of sensitivity calculations. A number of numerical examples illustrate the theoretical considerations.

Sensitivity Analysis, Numerical Computation, Constitutive Model, General Idea, Theoretical Consideration

The general problem of sizing, material and loading parameter sensitivity of non‐linear systems is presented. Both kinematic and path‐dependent material non‐linearities are considered; non‐linear sensitivity path is traced by an incremental solution strategy. The variational approach employed is quite general and can be employed for studying sensitivity of various path‐dependent highly non‐linear phenomena. Both the direct differentiation method (DDM) and adjoint system method (ASM) are discussed in the context of continuum and finite element mechanics. The merits of using the consistent tangent matrix and the necessity of accumulation of design derivatives of stresses and internal parameters are indicated. Aspects of sensitivity problems in metal forming are also discussed. A number of examples illustrate the paper.

Direct differentiation method, Adjoint system method, Metal forming

A complete mechanical model of lungs, valid for the analysis of stresses in tissues, which considers mechanical couplings between the tissue deformation and the air flow, is presented. The concept of a continuous description of the lung is adopted, with its volume assumed to be filled with a nonlinearly elastic, highly deformable solid - parenchyma, which is a thin-walled porous tissue structure on the micro-scale. Central airways are modelled as a discrete system of conduits immersed in the parenchyma. For the flow in peripheral airways and alveoli, a model of filtration in a three-dimensional two-phase porous medium (gas-tissue skeleton) is assumed. A new approach to pseudoelastic nonlinear constitutive modelling of parenchyma aiming at computational applications is presented. A fundamental set of nonlinear differential equations for the quasi-static coupled problem is derived and frictionless contact boundary conditions are included. A finite element formulation of the problem with an implicit time integration scheme is presented. The results of numerical examples obtained using the algorithm agree qualitatively with experimental data.

parenchyma

An interface finite element for non‐linear analysis of frictionless contact problems is presented. A constitutive equation relates stresses in the interface layer to the deformation gradient with respect to an imaginary reference configuration where the layer thickness is constant and finite. The equation of equilibrium is written in the same configuration, while the boundary conditions involve stresses related to an actual reference configuration used in the formulation for the contacting bodies. Finite element discretization is introduced for the layer in order to calculate the unknown field of displacements. Computational examples demonstrate properties of the contact element and its range of applicability in the analysis of large deformation contact problems including relative sliding of curved surfaces.

The anatomy and geometry of the lung at the micro- and macroscopic level have been described briefly. A notion of lung parenchyma — a macroscopically continuous medium whose mechanical properties result from those of microstructural components — has been adapted. Simplifying assumptions propounded in the constitutive model have been discussed. Two phases have been distinguished in the medium: the solid phase — a highly deformable, nonlinearly elastic skeleton in the form of a thin-walled tissue structure on the micro-scale — and the fluid phase — perfect gas (air) filterating through the structure. General constitutive relations for both phases and their mechanical interactions have ben formulated. Further, the fundamental set of differential equations of the quasi-static coupled problem has been developed. Large deformations, material nonlinearities, and dependence of permeability on skeleton deformation have been included. Matrix formulation of the problem has been presented from the point of view of the finite element method. An implicit iterative time integration scheme has been proposed. The algorithm has been illustrated with results of simple numerical tests.

The paper presents a review of constitutive models recently applied in the mechanical analysis of living tissues. The mechanical properties of the basic microstructural components of animal tissues (connective tissue fibers, hydroxyapatite, muscular cells) are discussed. Next, on the basis of available literature, a series of constitutive models for various tissues is presented. The models are classified in regard to the type of constitutive relations. Three basic groups of models are distinguished: viscoelastic, nonlinearly-elastic and linearly-elastic. Examples of constitutive equations are provided, followed by the discussion of their properties and range of applicability.

This paper presents modelling of both single and double-phase powder sintering processes at the macroscopic level. In particular, its constitutive formulation, numerical implementation and numerical tests are described. The macroscopic constitutive model is based on the assumption that the sintered material is a continuous medium. The parameters of the constitutive model for material under sintering are determined by simulation of sintering at the microscopic level using a micro-scale model. Numerical tests were carried out for a cylindrical specimen under hydrostatic and uniaxial pressure. Results of macroscopic analysis are compared against the microscopic model results. Moreover, numerical simulations are validated by comparison with experimental results. The simulations and preparation of the model are carried out by Abaqus FEA - a software for finite element analysis and computer-aided engineering. A mechanical model is defined by the user procedure “Vumat” which is developed by the first author in Fortran programming language. Modelling presented in the paper can be used to optimize and to better understand the process.

Educational assessment, Hydrostatics, Computer simulation, Finite-element analysis, Programming languages, Sintering

This paper presents modeling of a sintering process at various scales. Sintering is a powder metallurgy process consisting in consolidation of powder materials at elevated temperature but below the melting point. Sintering models at the atomistic, microscopic and macroscopic scales have been presented. Sintering is a process governed by diffusion therefore the atomistic modeling using the molecular dynamics has been focused on investigation of the diffusion process. The micromechanical model has been developed within the framework of the discrete element method. It allows us to consider microstructure and its changes during sintering. The macroscopic model is based on the continuum phenomenological approach. It combines elastic, thermal and viscous creep deformation. The methodology to determine macroscopic quantities: stress, strains and constitutive viscous properties from the discrete element simulations has been presented. Possibilities of the developed models have been demonstrated by applying them to simulation of sintering of the intermetallic NiAl powder. Own experimental results have been used to calibrate and validate numerical models.

sintering, modeling, discrete element method, diffusion, molecular dynamics, macroscopic model

W niniejszej pracy rozważamy problem zbieżności i jednoznaczności rozwiązania nieliniowego problemu symulacji procesów tłoczenia blach modelowanych z wykorzystaniem sztywno-lepkoplastycznego modelu materiału. Do symulacji wykorzystujemy programy MFP2D i MFP3D sprawdzone w kilku zastosowaniach przemysłowych. Zaletą obydwóch kodów numerycznych jest względna prostota algorytmu, a co za tym idzie możliwość dokładnej analizy wrażliwości i optymalizacji procesu tłoczenia blach. Niestety oba programy mają poważną wadę związaną z przyjętym modelem materiału. W trakcie obliczeń obserwujemy niestabilność wywołaną stosunkowo wysokim wskaźnikiem uwarunkowania macierzy układu równań. Dodatkowo przyjęty model kontaktu zawierający funkcję kary pogarsza wskaźnik uwarunkowania macierzy sztywności. Celem badań jest między innymi poprawienie tego wskaźnika.

tłoczenie blach, teoria plastycznego płynięcia, wskaźnik uwarunkowania macierzy

A computational model of adaptive bone remodelling is formulated as an optimization problem of instantaneous changes in microstructure that minimize a functional describing the structure quality rate. Microstructure is locally described by a set of scalar geometric parameters. Macroscopic (continuum) elastic properties are assumed anisotropic and expressed as known functions of the geometric parameters. Strain energy is considered as the quality measure of bone at given load conditions. Rate of geometric parameters is postulated to minimize the rate of the quality functional. An optimization problem is formulated in the continuum description and then iit is discretized both in space and time, taking finally a form of a system of nonlinear algebraic equations on unknown microstructure parameter increments, repeatedly solved in subsequent time instants.

adaptive remodelling, anisotropy, finite element method, optimization

Constitutive properties of cancellous bone depend on microstructural geometry. Evaluation of the interrelationship is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of ‘equivalent’ trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. The models are a geometrically enhanced version of those presented in the author’s previous work. Static numerical tests are performed with the finite element method for an exhaustive number of parameter value sets (microstructure instances). Repeatability of the microstructure allows to test only a sigle cell with appropriate boundary conditions. Values of computed stresses and strains allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.

Cancellous Bone, Equivalent Microstructure, Constitutive Modelling, Elastic Properties, Orthotropy, Fabric

Natural (or biological) materials generally achieve excellent mechanical performance. In particular, trabecular bone has a high stiffness / strength / weight ratio, which is why its structure inspires the development of new ultralight cellular materials. This work introduces a multiscale method for the design of elastic solids with a parametrized biomimetic microstructure of trabecular bone. The method combines a finite element model to evaluate the stiffness of the body at the macroscale with a non-linear optimizer to obtain the optimal values ​​of the microparameters and the orientation of the microstructure on the body domain. The most outstanding characteristics of its implementation are the use of: (i) an interior point type optimization algorithm, (ii) a precalculated response surface methodology for the evaluation of the elastic tensor of the microstructure as a function of the microparameters, ( iii) the attached method for calculating the sensitivity of the macroscopic mechanical response to the variation of the microparameters, and (iv) a weighted filter on the microparameters to correct for checkerboard effects. The performance and effectiveness of the tool are evaluated by solving a problem that consists of finding the optimal distribution of microstructures for the proximal end of a femur subjected to physiological loads. Three strategies are evaluated for the specification of the solid volume fraction constraints. The results are compared with data from a CT scan of a real human bone. The model successfully predicts the main features of the spatial arrangement of the trabecular and cortical microstructures of natural bone.

multiscale optimization, trabecular bone, parametric miscrostructure

multiscale structural optimization, biomimetic microstructures, finite element analysis

This paper presents modelling of double-phase powder sintering processes at the macroscopic level. In particular, its constitutive formulation, numerical implementation and numerical simulations are described. Numerical tests were carried out for a cylindrical specimen under uniaxial pressure and are compared against the microscopic model results. The model has been developed within the framework of a MUSINT project which is carried on in Institute of Fundamental Technological Research, Warsaw, Poland. The overall objective of the MUSINT (Multiscale numerical modelling of sintering processes) is development of numerical models allowing us to analyse at various scales manufacturing processes employing sintering as the main technological stage.

finite element method, sintering, multiscale modelling, double-phase composite

This document presents modelling of single-phase powder sintering processes at the macroscopic level. In particular, its constitutive formulation, numerical implementation and numerical test are described. Numerical tests were carried out for a cylindrical specimen under hydrostatic and uniaxial pressure. Results of macroscopic analysis are compared against the microscopic model results.

sintering porcesses, numerical analysis, multiscale modelling

sintering, discrete element method, multi-scale modeling

sintering, constitutive model

sintering, multiscale modelling

This paper presents NUMPRESS System that has been developed in IPPT PAN as a result of a project financially supported by European Regional Development Fund (within the Innovative Economy Programme) and is dedicated to small and middle enterprises dealing with sheet metal forming. It seems undoubted that efficient design of an industrial sheet forming process requires reliable computer simulations and a tool for numerical optimization of the process parameters. It has to be also admitted that. among small and medium enterprises (SME) in this industrial branch, there are many who do not use any such numerical tools in their practice.
Computer simulation of sheet metal forming processes is a very specific branch of computational mechanics. Finite element systems dedicated strictly to this kind of processes are needed and actually present on the market. Commercial systems (like Autoform, PAM-Stamp, Stampack, etc.) are, due to their prices, usually beyond financial ability of SME.

Design of the drawing process and tools, i.e. choice of proper values of several design parameters, require efficient optimization strategy. In this process, random character of at least some of the parameters has to be taken into account. In view of this fact, the traditional, deterministic approach to optimization is insufficient and elements of robust design optimization techniques and reliability analysis have to be included in the formulation of the optimization problem. It has to be admitted that, even if some of the mentioned commercial simulation systems offer numerical optimization modules, not all of them reach beyond the deterministic concept of the optimization process.

sheet metal forming, finite element method, deterministic and robust design optimization, reliability analysis

This paper presents basic features of the NUMPRESS system and some examples ofuse. The system has been developed at IPPT PAN as a result of a project financially supported by European Regional Development Fund and is dedicated to small and middle enterprises (SME) dealing with sheet metal forming. The program consists of (i) an analytical finite element method module (ii) an optimization module, (iii) a reliability analysis module, and (iv) a graphical user interface enabling communication between modules. The analytical module consists of two independent programs up to the user’s choice: NUMPRESS-Flow, a faster and less accurate program for implicit quasi-static analysis of rigid-viscoplastic shells (based on the flow approach) and NUMPRESS-Explicit, a program for explicit dynamical analysis of elastic-plastic shells. Both programs are interfaced to a well-known commercial graphical pre-and postprocessor GiD.

sheet metal forming, finite element method, deterministic and robust design optimization, reliability analysis

The objective of the paper is formulation of sensitivity analysis for flow approach simulations of deep drawing problems with respect to arbitrary design parameters. First, finite element formulation of the primary problem is presented. Its important feature is the full algorithmic tangent viscosity matrix which, as it will be shown, is a necessary tool in sensitivity calculations. The algorithmic (consistent) tangent matrix is unfortunately asymmetric, which is addressed in our considerations. The semi-analytical formulation of sensitivity is used, which means that some complex design derivatives in the sensitivity equations are estimated by finite difference method. The finite difference method will be used as a reference method for verification of the sensitivity results.

finite element method, flow approach, sensitivity analysis

finite element method, contact analysis, sheet metal forming

finite element method, contact analysis, sheet metal forming

An improved approach to penalty modeling in contact mechanics is proposed. The presented algorithm enables evaluation of the optimum values of the penalty factor for each constrained degree of freedom in the finite element model. The values are chosen so as to ensure desired accuracy in fulfillment of the geometric constraints while keeping the condition number of the modified stiffness matrix at a moderate level. Thus, the main weakness for which the penalty approach is often criticised — excessive worsening of the system conditioning—is fairly limited. Numerical examples confirm advantages of the method.

contact modeling, penalty approach, matrix condition number

In this study, a numerical model that allows to simulate the process of anisotropic remodelling of cancellous bone is presented. The bone is treated as continuum with linear elastic anisootropic mechanical properties that result from predefined microstructure topology and parameterized geometric proportions of trabeculae. Elastic constants and relative density are explicitly known functions of the geometric parameters. The microstructure parameters (including orientation directions) are non-uniformly distributed in the bone volume and subject to changes in time according to a remodelling rule that takes the form of minimization of the elastic strain energy rate. Special attention is focused on constraints in the optimization procedure that allow to eliminate unrealistic microstructure instances.

cancellous bone, constitutive model, orthotropy, bone remodelling

Uni-directional fiber–matrix composite, Parametric constitutive model, Finite element analysis, Homogenization, Representative volume element, Parameter sensitivity

A numerical model that allows to simulate the process of anisotropic remodelling of cancellous bone is presented. The bone is treated as continuum with linear elastic orthotropic mechanical properties. Elastic constants and relative density are explicitly known functions of geometric parameters of microstructure. The parameters are nonuniformly distributed in the bone volume. The remodelling rule is an optimization problem in which the “cost” functional is a time rate of a certain global measure of bone quality at a given load state. Instantaneous rates of the parameters are supposed to minimize the functional. The numerically predicted evolution of the parameters is obtained from the time integration of the results of the instantaneous optimization problem.

cancellous bone, constitutive model, orthotropy, bone remodelling

design sensitivity analysis, finite element method, shell structures, elasto-plasticity

Design sensitivity analysis, finite element method, finite rotations, shell elements

The paper describes a constitutive algorithm for elastic-plastic finite rotation shells and explicit dynamics with design derivatives calculated w.r.t. We show that despite a great complexity of the solution algorithm for the finite-rotation elastic-plastic shells, it is feasible to compute analytical design derivative of this algorithm, and the yielded sensitivities are of very good accuracy.

design sensitivity analysis, finite elment method, shell structures, dynamics, finite rotations