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Purpose The purpose of this study, which is designed for the implementation of models in the implicit finite element framework, is to propose a robust, stable and efficient explicit integration algorithm for rate-independent elasto-plastic constitutive models. Design/methodology/approach The proposed automatic substepping algorithm is founded on an explicit integration scheme. The estimation of the maximal subincrement size is based on the stability analysis. Findings In contrast to other explicit substepping schemes, the algorithm is self-correcting by definition and generates no cumulative drift. Although the integration proceeds with maximal possible subincrements, high level of accuracy is attained. Algorithmic tangent stiffness is calculated in explicit form and optionally no analytical second-order derivatives are needed. Research limitations/implications The algorithm is convenient for elasto-plastic constitutive models, described with an algebraic constraint and a set of differential equations. This covers a large family of materials in the field of metal plasticity, damage mechanics, etc. However, it cannot be directly used for a general material model, because the presented algorithm is convenient for solving a set of equations of a particular type. Practical implications The estimation of the maximal stable subincrement size is computationally cheap. All expressions in the algorithm are in explicit form, thus the implementation is simple and straightforward. The overall performance of the approach (i.e. accuracy, time consumption) is fully comparable with a default (built-in) ABAQUS/Standard algorithm. Originality/value The estimated maximal subincrement size enables the algorithm to be stable by definition. Subincrements are much larger than those in conventional substepping algorithms. No error control, error correction or local iterations are required even in the case of large increments.

Injection molded products, produced from semi-crystalline polymers, may include undercut features which can introduce distortion to the shape of the product during ejection. A thermo-mechanical modeling approach for simulating these advanced ejection problems is developed. The approach is formed by combining a method for three-dimensional residual stress prediction and an advanced material model for modeling the solid visco-elasto-plastic mechanical behavior. The task of this work is to assess, by analyzing a plaque-like product, the performance of the approach in the absence of the distortive ejection effects. The numerically predicted product shrinkage and mass at different packing pressure settings are compared to experimental results. The effect of packing pressure on product shrinkage and mass was reproduced by the model and the final residual stress field was found to be in accordance with the expectations. This confirms that the methodology could be used to analyze advanced ejection problems.

To obtain the initial conditions for ejection analysis of an injection molded part, a numerical simulation of the stress evolution in the material during injection molding is required. This topic, described in the literature only modestly for the full three-dimensional geometry, is addressed here by proposing an approach simple enough to be implemented in a general purpose solid mechanics simulation code. This feature makes it especially suitable with respect to the analysis of ejection, where custom code development might present hindering amount of additional work. As temperature and pressure field evolutions are obtainable through a computational fluid dynamics analysis, they are taken as input data in the stress analysis. The novelty of the approach is in the treatment of the melt region, where explicit tracking of the melt-solid interface is substituted by imposing the known pressure field in the melt region. The validity of the approach is experimentally tested by analyzing shrinkage and mass of moldings, as well as partial cavity pressure evolution at different packing pressure settings.

A fully coupled, partitioned, numerical model that accounts for fluid–structure interaction is applied for a study of the installation effects of Coriolis flowmeters. The modeled configurations include a single straight-tube full-bore flowmeter and two different twin tube flowmeters with straight and U-shaped measuring tubes. Three different flow disturbance elements positioned upstream of the flowmeter are considered in the study, as well as two different types of flow splitters in the case of the twin tube configurations. The installation effects are estimated by comparing the mass-flow sensitivities under the disturbed and fully developed flow conditions at the inlet of the flowmeter. For the modeled twin tube flowmeters they are found to be of the order of one-tenth of a per cent. These relatively small values of the installation effects are related to the presence of flow splitters and to the averaging of the motion of both measuring tubes in the twin tube configurations. Similarly, averaging the response from two sensor pairs instead of only a single sensor pair reduces the circumferential variations and the peak values of the installation effects for asymmetric flows in the single straight-tube flowmeter.

The paper presents a modelling of the laser hardening process by a high-power diode laser (HPDL). Through numerical implementation into the finite element method (FEM) code ABAQUS, the model is used in the computer simulation of two case studies of laser hardening selected for experimental validation. In the experiment, 100 × 100 × 15 mm cuboid samples made of 50CrV4 steel were subjected to laser hardening with significantly different sets of applied technological parameters (laser beam power, laser beam velocity) but still aiming at attaining a comparable maximum temperature on the sample surface. The simulation considers two alternative approaches to microstructure evolution and subsequent material hardness determination: one relying on the heating rate dependent austenitisation temperatures ( Ac 1 and Ac 3 ) governing microstructure transformation kinetics and the other neglecting heating rate dependence. Physical objectivity of the computed results is verified based on the corresponding temperature field measurements on the sample surface during heat treatment process and hardness measurements through the thickness of the laser-hardened sample. The experimental validation clearly proves that considering austenite kinetics at a high temperature change rate in computer simulation is definitely more physically congruent. In the study of the applied process parameters impact, the effect of a higher temperature change rate on austenite kinetics is shown by the temperature shift of austenite and ferrite to austenite start formations. From the investigation of the effect of different heat inputs providing the same maximum temperature on the sample surface it results that deeper area of increased hardness is established when less laser beam power and velocity are applied.

Ovaj rad obrađuje problem predviđanja mehaničkog odziva dijela zajedničke arterije glave (ZAG). Zaostala naprezanja u arteriji uzeta su u obzir uporabom termo-mehaničke analogije (TMA), koja se primjenjuje u ovom radu za potrebe modeliranja mehaničkog odziva dvoslojne strukture stijenke arterije. Obično se modeliranju zaostalih naprezanja u arteriji pristupi s uzdužno otvorenim modelom arterije, tzv. cut-open section, koji u slučaju bolesnikove arterije nije poznat. S TMA pristupom, umjesto uporabe uzdužno prerezane stijenke arterije, koja doduše osigurava početno stanje bez zaostalih naprezanja, u termo-mehaničkom modelu zajedničke arterije glave također je postignuto početno stanje bez zaostalih naprezanja ali na modelu stvarne, in vivo arterije. Tim pristupom, zaostalo naprezanje u ZAG aproksimirano je podvrgavanjem ZAG modela obujamskom deformacijom, t.j. primjenom odgovarajućih termičkih dilatacija. Takav pristup potvrđen je na modelu arterije kružnog presjeka i postizanjem stanja bez naprezanja u slučaju uzdužnog rezanja stijenke., This work deals with the prediction of the mechanical response of a section of a human common carotid artery (CCA). The arterial residual stress state is accounted for using the thermomechanical analogy (TMA) approach, which is applied in this work to model the mechanical response of a two-layered arterial structure. The starting point to model the arterial residual stress state is normally the cut-open section, which is in the case of patient-specific artery not known. With TMA approach, however, instead of using the arterial zero-stress cut-open configuration to predict the arterial residual stress state, a thermomechanical model of the CCA is considered with its zero-stress geometry defined based on the actual CCA in vivo configuration. The approximation to the CCA residual stress state is then obtained by exposing the auxiliary CCA model to a volumetric deformation, enforced via adequate thermal dilatations. The approach is validated on a circular arterial model and by predicting the CCA cut-open zero-stress state.

In this paper, an enhanced numerical method for forming tool design optimisation in three-dimensional (3D) sheet metal forming applications is presented. The applied procedure enables a determination of appropriate forming tool geometry so that the manufacture of a sheet metal product inside specified tolerances would be ensured. In addition to the springback that occurs in the formed part after removal of the forming tools, the impact of the thinning of the sheet metal during the forming process is considered in the method, and both effects are correspondingly compensated for an iterative procedure. Computational efficiency in the E-DA-3D method is achieved mainly because the improved accuracy of the communicated data established corresponding interrelations between the discretised topologies used in the definition of the prescribed product geometry, the current tool geometry, and on this basis actually computed product geometry which is achieved by means of additional point topology mappings. The potential and effectiveness of the method is demonstrated by considering two cases of the forming tool design optimisation that are also experimentally validated.

The paper deals with constitutive modeling of highly anisotropic sheet metals and presents FEM based earing predictions in a round cup drawing simulation of highly anisotropic aluminum alloys where more than four ears occur. For that purpose the BBC2008 yield criterion, which is a plane-stress yield criterion formulated in the form of a finite series, is used. Thus defined criterion can be expanded to retain more or less terms, depending on the amount of given experimental data. To be used in sheet metal forming simulations the constitutive model, derived in accordance with the associated flow theory of plasticity, has been implemented in a general purpose finite element code ABAQUS/Explicit via VUMAT subroutine, considering alternatively different number of parameters in the BBC2008 yield criterion, where possible number of parameters are any multiple of number 8. For the integration of the constitutive model the explicit NICE (Next Increment Corrects Error) integration scheme has been used. The CPU time consumption for an explicit deep drawing simulation, which is based on the developed constitutive model, has been proven to be, due to effectiveness of the used integration scheme, fully comparable to the performance experienced when the simulation is performed with ABAQUS built-in constitutive models and implicit integration schemes. Two aluminum alloys, namely AA5042-H2 and AA2090-T3, have been considered for a validation of the constitutive model. The respective BBC2008 model parameters have been identified for both alloys with a developed numerical procedure, based on a minimization of the specified cost function. For both materials, the simulation predictions based on the BBC2008 model prove to be in very good agreement with the experimental results. Further, in order to show the flexibility of the BBC2008 model in modeling of highly anisotropic sheet metal response, we have introduced a highly anisotropic fictitious material which yields, according to the theory, twelve ears in cup drawing. As it is shown in the paper the BBC2008 model is able to predict twelve ears in cup drawing simulation with the formulation containing 16 parameters for anisotropy description only. The flexibility and accuracy of the constitutive model together with the robust identification and integration procedure guarantee the applicability of the BBC2008 yield criterion in industrial applications.

The paper presents a derivation of the consistent tangent operator (CTO) for the cutting-plane algorithm (CPA). For a class of plasticity models that are suitable to be integrated using CPA, an explicit recursive expression is analytically derived and is updated in each iteration of the CPA integration procedure to yield the final value of the CTO when the CPA is converged.

The paper focuses on the modelling of springback within a formed stainless steel sheet. The main subject of this work is the construction of aconstitutive model which simultaneously considers sheet anisotropy, damage evolution, and stiffness degradation in material during forming.The developed model is based on the Gurson-Tvergaard-Needleman damage model, which is adequately extended by the implementationof the anisotropic Hill48 plasticity and MoriTanakas approach to stiffness degradation. Considering the established relationships, somematerial parameters that are included in the model are characterised by the corresponding measurements. The experimental validation ofthe developed constitutive model is performed on a springback test, which consists of bending and releasing rectangular stainless steelspecimens that were previously plastically prestrained to a different degree, either in the rolling or transverse direction. A comparison ofthe proposed modelling approach to the classical approach by using the Hill48 model clearly indicates that the simultaneous modelling ofmaterial phenomena, especially the coupling of stiffness degradation with anisotropic plasticity, can be the true key to obtaining a moreaccurate prediction of the springback in sheet-metal-forming applications. Članek obravnava konstitutivno in numerično modeliranje procesa preoblikovanja nerjavne pločevine s poudarkom na napovedi elastične povračljivosti. Znano je, da se med obremenjevanjem v pločevini poleg trajne deformacije razvija tudi poškodovanost v obliki mikropraznin, votlinic in razpok. Pojav je razpoznaven z opazovanjem mikrostrukture pločevine, ki je bila predhodno obremenjena do določene stopnje trajne deformacije. Glede na teorijo mehanike poškodb so tako nastale praznine in mikro razpoke glavni vzrok za degradacijo togosti v duktilnih materialih, kar ima nadalje pomembno vlogo pri napovedovanju elastične povračljivosti. Ker je pločevina zaradi procesa izdelave anizotropna, ima tudi to pri napovedovanju pomemben vpliv.

The response of the hypo-eutectoid steel to laser hardening, which is measured as the depth of the increased hardness, depends not only on the set of the process parameters but also on the prior microstructure of the workpiece. The multiple preliminary stages of the treatment of the workpiece in the industrial conditions are commonly not completely known, resulting in an unclear prior microstructure of the workpiece. To model the response of the hypo-eutectoid steel, a validated numerical model for laser hardening has been used in the computer simulation of the process for four different cases. The numerical model takes into account the 3D geometry of the workpiece, its prior microstructure, and the effect of the heating rate during the laser hardening process on the kinetics of the phase transformation. The four cases were designed to take into account two different sets of process parameters and two different prior microstructures of the workpiece. The output of the computer simulation was verified experimentally.

The purpose of this work is to predict the effect of impaired red blood cells (RBCs) deformability on blood flow conditions in human carotid artery bifurcation. First, a blood viscosity model is developed that predicts the steady-state blood viscosity as a function of shear rate, plasma viscosity, and mechanical (and geometrical) properties of RBC's. Viscosity model is developed by modifying the well-known Krieger and Dougherty equation for monodisperse suspensions by using the dimensional analysis approach. With the approach, we manage to account for the microscopic properties of RBC's, such as their deformability, in the macroscopic behavior of blood via blood viscosity. In the second part of the paper, the deduced viscosity model is used to numerically predict blood flow conditions in human carotid artery bifurcation. Simulations are performed for different values of RBC's deformability and analyzed by investigating parameters, such as the temporal mean wall shear stress (WSS), oscillatory shear index (OSI), and mean temporal gradient of WSS. The analyses show that the decrease of RBC's deformability decrease the regions of low WSS (i.e., sites known to be prevalent at atherosclerosis-prone regions); increase, in average, the value of WSS along the artery; and decrease the areas of high OSI. These observations provide an insight into the influence of blood's microscopic properties, such as the deformability of RBC's, on hemodynamics in larger arteries and their influence on parameters that are known to play a role in the initiation and progression of atherosclerosis.

In the paper, we present an enhanced numerical method for the forming tool design optimisation in sheet metal forming. The applied procedures enable a determination of appropriate forming tool geometry so that manufacture of a formed product with specified geometry would be ensured. Apart from springback occurred by the formed part after removal of the forming tools also impact of thinning of the sheet metal during the forming process is considered in the method, and both effects are correspondingly compensated in an iterative procedure of the forming tool geometry determination. The enhanced displacement adjustment method (E-DA) is based on the well-known displacement adjustment (DA) method the application of which is indeed relatively simple, but has proved also, due to an increased number of iterations needed to achieve the required tolerance and possible loss of accuracy, to be less successful when forming of parts with more complex geometry is considered. Computational efficiency in the E-DA method is achieved by applying additional point topology mappings, which establish corresponding interrelations between the discretised point topologies used in the definition of the prescribed product geometry, current tool geometry and on this basis actually computed product geometry, contributing thus significantly in improving the accuracy of communicated data. The advantage of the improved method over the conventional DA method is demonstrated by considering the forming tool design optimisation in channel bending and forming of cylindrically symmetric products.

The article introduces, as a result of further development of the first-order scheme NICE, a simple and efficient higher-order explicit numerical scheme for the integration of a system of ordinary differential equations which is constrained by an algebraic condition (DAE). The scheme is based on the truncated Taylor expansion of the constraint equation with order h of the scheme being determined by the highest exponent in the truncated Taylor series. The integration scheme thus conceived will be named NICEh, considering both principal premises of its construction. In conjunction with a direct solution technique used to solve the boundary value problem, the NICEh scheme is very convenient for integrating constitutive models in plasticity. The plasticity models are defined mostly by a system of algebraic and differential equations in which the yield criterion represents the constraint condition. To study the properties of the new integration scheme, which, like the forward-Euler scheme, is characterised by its implementation simplicity due to the explicitness of its formulations, a damage constitutive model (Gurson–Tvergaard–Needleman model) is considered. The general opinion that the implicit backward-Euler scheme is much more accurate than the thus-far known explicit schemes is challenged by the introduction of the NICEh scheme. The accuracy of the higher-order explicit scheme in the studied cases is significantly higher than the accuracy of the classical backward-Euler scheme, if we compare them under the condition of a similar CPU time consumption.

Springback, a phenomenon that is governed by elastic strain recovery after the removal of forming loads, is of great concern in sheet metal forming. There is no doubt that in this regard, physically reliable numerical modelling of the forming process and predictions of springback obtained by respective computer simulations are crucial for controlling this problem. Unfortunately, by currently available approaches, springback still cannot be adequately predicted in general. In this paper, a new constitutive model is proposed which considers simultaneously sheet anisotropy, damage evolution and strain path-dependent stiffness degradation during sheet metal forming. For parameter identification of the built constitutive model, a particular experimental procedure is developed and an optimization procedure is employed to solve the inverse problem that arises. The proposed approach to constitutive modelling is validated in the end by a simulation of the springback in the formed HSS steel sheet. The simulation results, which prove to be in good agreement with the experimental ones, lead to the conclusion that accurate modelling only of anisotropic yielding is not enough to accurately predict the springback phenomenon; the constitutive model should also include the strain path-dependent change of the elastic moduli.

In accordance with the great importance given to the subject of stiffness degradation, in particular with regard to metal forming, this work experimentally investigates the anisotropic elastic properties of plastically prestrained cold-rolled sheet metal (stainless steel EN 1.4301, also AISI 304). From the experiments performed, two main conclusions regarding stiffness degradation can be extracted. First, since under specific stretching the intensity of the normalized Young’s moduli degradation in both directions remains approximately similar, it may be concluded that the potential initial elastic anisotropy tends to be preserved during loading. Second, as the evidenced stiffness degradation has proved to be strongly correlated with the stretching direction of the sheet metal, it can be concluded that the stiffness evolution in the cold rolled sheet steel is path dependent. These interesting discoveries also provide some answers for modelling the kinetic damage evolution laws in damage mechanics.

Precise integration of a constitutive law is required whenever a boundary value problem is to be solved. In this paper a new explicit scheme for the integration of elasto-plastic constitutive models, that consist of algebraic yield condition and differential evolution equations, is presented. The scheme was primarily developed for the integration of constitutive models in boundary value problems which are solved with a direct solving technique (like in Abaqus/Explicit). Although the scheme is as fast and as simple for implementation as unstable explicit forward-Euler scheme, it can reach the same level of accuracy as commonly used backward-Euler scheme. Two examples are shown in the paper, the first one being simple for demonstration purposes only, whereas in the second one the scheme is also implemented in the Abaqus/Explicit user subroutine VUMAT. The obtained results confirm that the scheme is accurate and stable. It can be concluded in consequence, that the scheme is very handy for implementation of complex elasto-plastic constitutive laws into finite element software.

The paper presents a simple but efficient new numerical scheme for the integration of nonlinear constitutive equations. Although it can be used for the integration of a system of algebraic and differential equations in general, the scheme is primarily developed for use with the direct solution methods for solving boundary value problems, e.g. explicit dynamic analysis in ABAQUS/Explicit. In the developed explicit scheme, where no iteration is required, the implementation simplicity of the forward-Euler scheme and the accuracy of the backward-Euler scheme are successfully combined. The properties of the proposed NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, are compared with the properties of the classical forward-Euler scheme and backward-Euler scheme. For this purpose two highly nonlinear examples, with the von Mises and GTN material model considered, have been studied. The accuracy of the new scheme is demonstrated to be at least of the same level as experienced by the backward-Euler scheme, if we compare them on the condition of the same CPU time consumption. Besides, the simplicity of the NICE scheme, which is due to implementation similarity with the classical forward-Euler scheme, is its great Advantage.

The article presents a simple but efficient numerical scheme for the integration of non-linear constitutive equations, in which the principal reason for the inaccuracy of the classical explicit schemes, for example forward-Euler scheme, is effectively eliminated. In the newly developed explicit scheme, where there is no need for iteration, the implementation simplicity of the forward-Euler scheme and accuracy of the approach, which is using the backward-Euler scheme to integrate the constitutive equations, are successfully combined. Computational performance of the proposed next increment corrects error (NICE) integration scheme, particularly regarding the accuracy and the CPU time consumption, is first analysed on a case of complex loading of a material point. When comparing it to the forward-Euler, backward-Euler, trapezoidal and midpoint integration schemes, it turns out that because of its capability of a fast and relatively accurate integration of the constitutive equations, the NICE scheme is very convenient for the integration of constitutive models, where a direct solution technique is used to solve a boundary value problem. Although the deduction of the new integration scheme is general, its implementation for shell applications needs particular care. Namely, in order to satisfy the zero normal stress condition during the whole integration, a through-thickness strain increment has to be adequately chosen in each integration step. The NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, has been additionally compared with the ABAQUS/Explicit default integration scheme (backward-Euler) and forward-Euler scheme. Two loading case-studies, namely the bending of a square plate and the stretching of a specimen including the onset of necking, are considered with two constitutive models—the von Mises and GTN material model being adopted. Generally, the NICE scheme has demonstrated to be advantageous in cases, where reasonable accuracy and very fast integration of the constitutive model is demanded, which is mostly the case in engineering computations with a direct solution method, for example explicit dynamics and metal forming process simulations. Copyright © 2009 John Wiley & Sons, Ltd.

The metal sheet fine blanking process of a circular hole is simulated using a two-dimensional axisymmetric finite element model. A numerical model used in our computer simulation is based on a dynamic explicit scheme. Yield curve that is implemented in the numerical model is obtained upon tension test data. Its particularity is first the extension into a post-necking part, which is determined by an inverse identification procedure, and second, the abrupt reduction to a small constant value after the maximal equivalent plastic strain is reached. To reduce finite element mesh distortion in the sheared zone remeshing steps, based on arbitrary Lagrangian-Eulerian formulation, are implemented during the simulation. After the apparent crack is developed through the metal sheet, the sheet domain is divided into two domains with possible friction contact evolution between the separated domains enabled. Taking contact evolution between the separated domains into account results in a better determination of the final fine blanked surface. Time evolution of the punch force obtained in computer simulation proves good agreement with the experimental one.

Springback, the phenomenon that is governed by strain recovery after removal of forming loads, is of great concern in sheet metal forming. There is no doubt that physically reliable numerical modelling of the forming process and predictions of springback obtained by respective computer simulations are crucial to control this problem. Unfortunately, by currently available approaches springback still cannot be adequately predicted in general. This paper is an attempt of building a corresponding constitutive model, which will simultaneously consider sheet anisotropy, damage evolution and strain path dependent stiffness degradation during metal forming. First, for the identification of the parameters in the built constitutive model a particular experimental procedure is deliberately developed. To solve the arisen inverse problem an optimization procedure is employed. The proposed approach to constitutive modelling is validated in the end by a simulation of the springback in the formed HSS steel. The simulation results prove to be in good agreement with the experimental ones. From the performed comparisons it is clearly indicated, that only simultaneous modelling of material properties can be the true key to obtain accurate prediction of springback in sheet metal forming.

Deflection analysis of beams with rectangular cross section is considered under specific loading conditions, resulting in at most quadratic bending moment distribution, and assuming elasto-plastic behaviour with no hardening. Within the framework of small strain and small displacement approach analytical solutions are derived, which enable elasto-plastic analyses of beams to be performed in a closed analytical form. In consequence, clear tracing of the elasto-plastic response evolution with a propagation of the plastic zone through the volume, i.e. its spreading along the beam’s longitudinal axis as well as its penetration through the cross section, is enabled as loads increase, from the appearance of a first plastic yielding in a structure till its collapse. With the derivation of the general solution, listed explicitly by Eqs. (21)–(23), which was never presented in any article or book before, the presented article fills the gap in the analytical non-linear mechanics of beams.

The aim of this work is to show first, how the springback of a steel sheet drawn part is affected by the stiffness degradation, as it results from the damage evolved during forming process, and second, to build a respective modeling approach to take this degradation into account. For the consideration of the orthotropic elastic properties degradation we develop an approach, based on the Mori-Tanaka theory, where damage is considered by inclusion of ellipsoidal cavities. The respective void shape evolution is proposed to be identified with the measurements of elastic modulus in two perpendicular directions during the uniaxial tensile test of a flat specimen at different loading stages. The proposed approach is coupled with the Gurson-Tvergaard-Needleman (GTN) plastic potential, though it could be substituted by almost any other continuum damage model. At the end the presented approach is experimentally validated by a simple springback test, developed by authors. A very good agreement between by calculation predicted and measured springback amount is observed.

The paper presents a coupled numerical model built to simulate the operation of Coriolis flowmeters, which exploit the alteration of the vibration mode shape of the measuring tube for the mass flow rate measurement. The explained measuring effect is a consequence of the interaction between the motion of the tube, vibrating at its natural frequency, and the fluid flow in it. The numerical model is realized by coupling of a finite volume (FV) code for fluid flow analysis with a finite element (FE) code for structural analysis using the conventional staggered solution procedure, with added inner iterations to achieve strong coupling. The simulation algorithm is divided into two steps. A free vibration of the measuring tube considered in the first step is complemented in the second step, after the numerical free vibration response is properly stabilized, with the harmonic excitation force actuating the measuring tube at its resonant frequency of several hundreds of Hertz to resemble the operation of actual Coriolis flowmeters. Different scenarios using zero-order or three-point fluid load predictor and soft application of the fluid load in the initial stages of the simulation are compared to yield a simulation strategy, which will minimize the time needed to obtain the stabilized steady-state response of the vibrating measuring tube. The proposed simulation procedure was applied on a straight-tube Coriolis flowmeter and used for the estimation of the velocity profile effect. The results exhibit sufficient stability (low scatter) to be used for the estimation of sensitivity variations of order of magnitude around tenths of a percent.

The prediction of the arterial zero-stress state is as of yet an unresolved problem in the field of modelling the mechanical response of patientspecific arteries. This is because the configuration associated with arterial zero-stress state is impossible to obtain experimentally. However, the zero-stress configuration (or, equivalently, the residual stresses related to this configuration) represents the crucial data of any numerical analysis. In this study, the mechanical response of a pre-stressed, pressurised, hyperelastic tube (representing the artery) is determined without knowing the initial zero-stress configuration of the artery. Instead, to predict the arterial residual (bending and stretching) stresses, a corresponding thermomechanical analogy is used. As shown in the paper, the arterial residual stress state can equally be obtained by thermally loading a properly defined closed tube. Thus, based on the loaded state of the corresponding thermomechanical model, the arterial residual stress sate is constructed, from where the arterial loaded state can be obtained. The initial configuration of the thermomechanical model is defined on the basis of the arterial loaded configuration. The methodology is validated by predicting the zero-stress state of the artery. The predicted equilibrium state of the artery, when cut longitudinally and transversally, has the form of an opened-up tube with a relatively low stress state in comparison to the arterial residual stresses. The results thus demonstrate that arterial residual stresses that are predicted with the corresponding thermomechanical model exhibited the bending distribution, which proves the methodology to be adequate. Določitev breznapetostnega stanja arterij predstavlja še vedno nerazrešen problem na področju numeričnega modeliranja mehanskega odziva realnih arterij. Breznapetostno stanje arterije, ki ima v splošnem obliko krožnega sektorja, je namreč eksperimentalno nemogoče določiti. Oblika breznapetostnega stanja (oziroma temu pripadajoče zaostalo napetostno stanje) pa je pomembno zato, ker predstavlja začetno stanje in tako ključen podatek sleherne numerične analize. Arterije se za karakterizacijo njihovega mehanskega odziva običajno obravnava kot idealno krožne cevi, to pomeni idealno krožne in ravne. Takšna poenostavitev omogoča določitev zaostalih napetosti arterije brez poznavanja njihovega breznapetostnega stanja. Prisotnost zaostalih napetosti je posledica homogenizacije odziva arterij in vivo. Napetostno stanje arterije in vivo bi bilo namreč izrazito nehomogeno z velikim transmuralnim gradientom brez upoštevanja zaostalih napetosti. Zaradi njihove porazdelitve skozi debelino žilne stene se jih običajno asociira z upogibnimi napetostmi. Na notranji površini kolobarja so zaostale napetosti tlačne, medtem ko so na zunanji površini natezne.

A coupled finite-volume (FV)/finite-element (FE) numerical model of the straight-tube Coriolis flowmeter is considered. It uses the staggered partitioned algorithm with additional pressure predictor and interfield iterations to minimize the time lag in the fluid–structure coupling procedure. The solutions were evaluated in terms of the fundamental natural frequency of the vibrating system and the corresponding phase difference between the motion of the symmetrically located sensing points on the measuring tube, which are actually exploited as the measuring effects of the Coriolis flowmeter for the fluid density and the mass flowrate, respectively. The FV/FE numerical model was validated by comparison with the solutions of the Euler beam and one-dimensional flow model, as well as with the solutions of the Flugge shell and potential flow model. The respective simulations were performed for different lengths of the measuring tube.

PurposeTo present numerical approaches to the solution of physically coupled non‐linear problems, which frequently happen to be characterized by their multi‐domain character.Design/methodology/approachBy adopting coupled solution strategies a considerable attention is devoted, in order to obtain a computationally efficient numerical algorithm, to the selection of appropriate space and time discretization, as well as to a proper discrete approximation method used.FindingsCoupling of two methods, the finite element method and the boundary element method, respectively, has proved to be computationally exceedingly advantageous, particularly in case of moving domains.Practical implicationsAs specific case studies computer simulation of an induction heating problem and a mushy‐state forming problem are considered. A thorough discussion on the coupling effects, characterizing the evolutions of respective physical quantities' fields, is given, and their impact on those evolutions is identified.Originality/valueThis paper presents efficient numerical strategies for the solution of a certain class of multi‐physics and multi‐domain problems.

By the computer-aided method of material characterization many complex identification cases, where classic experimental–analytical methods of physical properties identification fail, can be successfully solved. Such an example is the standard tensile test of a flat steel sample, where the yield curve cannot be identified after the occurrence of the necking phenomenon. Yet, in deep drawing of metal sheets, strains and stresses beyond the limits derived by classic analytic expressions upon the tensile test measurements are often met. In order to enable physically objective numerical simulations of those processes, reliable material properties data must be provided. To cope with the problem of the extended yield curve identification, a special combined experimental/numerical technique has been developed. The technique relies on the comparison between the real material response, measured by the standard tensile test, and the response, obtained from a numerical simulation of the same test under assumption of a prescribed material behaviour. By proper tuning of some characteristic parameters of this tentative material behaviour law, the numerical response can be drawn close to the measured one. For this purpose a special numerical approach, based on mathematical optimization methods is employed.

Summary An inverse identification method for characterization of wood sorptive properties is presented. The method relies on a computer simulation of a real experiment, in our case a desorption experiment, where spruce heartwood samples were dried from 27% to 8% moisture content. Three samples, distinguished by the respective moisture flow pattern through the specimen, were investigated. A computer aided material characterization using the so-called inverse problem identification method was performed on the measurements. The solution of the specified inverse problem enabled us to estimate the moisture diffusion coefficients of wood and to determine the moisture content field in the sample simultaneously. The method is first verified on two simple cases of uniaxial moisture flow, and then is used to characterize the diffusion coefficients on a biaxial moisture flow sample. In the latter case some salient features of the proposed method are exhibited.

Influence of a discontinuous nature of the elastoplastic systems response at large strain onto their sensitivity with respect to a design parameter is considered in the paper. It is discussed in the framework of the finite element modelling using the direct differentiation method for the sensitivity response calculation. Elastoplastic behaviour is formulated on the additive approach of the rate of deformation tensor and a hypoelastic characterization of the elastic response. It is shown that the computed sensitivity response of the system’s FE models can experience steep jumps on its overall path. This fact is confirmed by some examples.

In the present work, the parallelization of the solution of a system of linear equations, in the framework of finite element computational analyses, is dealt with. As the substructuring method is used, the basic idea refers to a way of decomposing the considered spatial domain, discretized by the finite elements, into a finite set of non-overlapping subdomains, each assigned to an individual processor and computationally analysed in parallel. Considering the fact that Personal Computers and Work Stations are still the most frequently used computers, a parallel computational platform can be built by connecting the available computers into a computer network. The incorporated computers being usually of different computational power and memory size, the efficiency of parallel computations on such a heterogeneous distributed system depends mainly on proper load balance. To cope the balance problem, an algorithm for the efficient load balance for structured and free 2D quadrilateral finite element meshes based on the rearrangement of elements among respective subdomains, has been developed.

In this paper a numerical approach to the process parameter study of the mechanical response of an aluminium heat exchanger during its manufacture is presented. The main motivation and the basic features of the numerical procedure used in this study are briefly described. Some results obtained by computer simulation of the manufacturing operation, as well as those obtained by experimental testing, are also presented.

The main benefit in using optimally shaped blanks in sheet metal forming is in maximizing the efficiency of the forming process and, since there is no need for additional cutting operations after the finished forming operation, in substantial reduction of the overall production cost. The paper presents a numerical method for optimal blank shape determination which is suitable in various sheet metal forming applications. The optimal blank shape is determined in iterative way so that the edge geometry of the formed product fits its reference geometry as close as possible. The iterative process starts at blank shape from which the product is produced with its edge fitting its reference geometry just approximately. In subsequent iterations the blank shape is continuously improved in accordance with the developed optimisation method. In order to determine the product edge geometry resulting from the current blank shape, a computer simulation of the forming process and the springback is performed in each iteration. Since its effectiveness highly depends on the quality and physical objectivity of the computer simulation, the developed numerical blank shape optimisation procedure has been validated also experimentally by considering forming of a product with rather complex edge geometry as the case study.

U ovom radu opisane su mehaničke analize primarnog cjevovoda Nuklearne elektrane Krško, koje obrađuju problem djelomično onemogućene termičke ekspanzije cjevovoda uslijed nedovoljnog razmaka među pločama ograničavača pomaka cijevi. Smanjenje razmaka između cijevi i pripadajućeg oslonca spriječava slobodno termičko proširenje cjevovoda tijekom zagrijavanja sistema. Time se prouzrokuju jake tlačne sile između cijevi i oslonaca, što se rezultira kao nepovoljno stanje deformacija i naprezanja i u cjevovodu i u potporama. Izvedene su serije numeričkih simulacija, koje potvrđuju dovoljnu čvrstoću primarnog cjevovoda Nuklearne elektrane Krško. Međutim, oslonci su puno više osjetljivi na opterećenja te vrste i stoga je za njih potrebno izvesti brižljivo konstrukcijsko razmatranje., Mechanical analyses of the Krško NPP primary system piping subjected to the restrained thermal expansion due to insufficient pipe shim clearances are addressed in the paper. Reduction of clearance between a pipe and respective pipe restraints preclude free thermal expansion of the pipe system during the heat-up, thus causing a rise of excessive contact forces in the pipe - pipe restraint pairs. This results in a development of unfavourable strains and stresses both in pipes and restraints. A series of numerical simulations was performed which demonstrate sufficient strength resistance of the Krško NPP primary piping, while supports are much more susceptible and need careful engineering consideration.

This paper presents a methodology based on the NICE integration scheme [1, 2] for simple and rapid numerical implementation of a class of plasticity constitutive models. In this regard, an algorithm is purposely developed for the implementation of newly developed advanced constitutive models into explicit finite element framework. The methodology follows the organization of the problem state variables into an extended form, which allows the constitutive models' equations to be organized in such a way, that the algorithm can be optionally extended with minimal effort to integrate also evolution equations related to a description of other specific phenomena, such as damage, distortional hardening, phase transitions, degradation etc. To confirm simplicity of the program implementation, computational robustness, effectiveness and improved accuracy of the implemented integration algorithm, a deep drawing simulation of the cylindrical cup is considered as the case study, performed in ABAQUS/Explicit. As a fairly complex considered model, the YLD2004-18p model [3, 4] is first implemented via external subroutine VUMAT. Further, to give additional proof of the simplicity of the proposed methodology, a combination of the YLD2004-18p model and Gurson-Tvergaard-Needleman model (GTN) is considered. As demonstrated, the implementation is really obtained in a very simple way.

Optimization of single and multistep wire drawing processes, based on the finite element analysis and nonlinear mathematical programming techniques, is considered. Since the investigated problem involves the optimal selection of the extrusion die angles and the correspondent reduction rate sequence, it can be considered as one belonging to a class of optimum shape design problems. Characteristic to the considered problem is that the domain boundary, the shape of which is to be optimized, is subject to the shape dependent forming loads. The minimization of the objective function, which is related to the process energy consumption, is sought by taking several technological and geometrical constraints into account. The numerically evaluated examples demonstrate that by using the proposed optimization technique not only a substantial cost reduction, but also a considerable improvement in the homogeneity of plastic deformation through the cross-section of the drawn wire can be achieved.

A computational strategy for an effective solution of a multibody contact problem with friction in the mechanics of deformable bodies is considered. Due to the coupling of quantities appertaining to the material points at the common contact boundary and due to imposing of additional constraints regarding the stress-displacement compatibility in the contact area, the primary set of the nonrelated one body system equations that is obtained according to the assumption of the nonconnected multibody system, is additionally augmented when a contact problem is considered. Due to the incorporated nonlinearities the analysis of contact problems requires a proper treatment of the history of the applied load. Therefore, to allow the external load to be applied in small finite steps the solution is traced in an incremental way, with the magnitude of the load increment being monitored by the contact compatibility. Namely, a reliable tracing of the contact compatibility variation along the application of the external load is achieved by introducing only one change of the contact status in each load increment. However, to fulfill all compatibility and contact constraints the incremental solution technique has to be additionally implemented by an iterative procedure within each increment, thus providing a means for satisfactory fulfilment of the contact compatibility.

The paper deals with constitutive modelling of highly anisotropic sheet metals. It presents FEM based earing predictions in cup drawing simulation of highly anisotropic aluminium alloys where more than four ears occur. For that purpose the BBC2008 yield criterion, which is a plane‐stress yield criterion formulated in the form of a finite series, is used. Thus defined criterion can be expanded to retain more or less terms, depending on the amount of given experimental data. In order to use the model in sheet metal forming simulations we have implemented it in a general purpose finite element code ABAQUS/Explicit via VUMAT subroutine, considering alternatively eight or sixteen parameters (8p and 16p version). For the integration of the constitutive model the explicit NICE (Next Increment Corrects Error) integration scheme has been used. Due to the scheme effectiveness the CPU time consumption for a simulation is comparable to the time consumption of built‐in constitutive models. Two aluminium alloys, namely AA5042‐H2 and AA2090‐T3, have been used for a validation of the model. For both alloys the parameters of the BBC2008 model have been identified with a developed numerical procedure, based on a minimization of the developed cost function. For both materials, the predictions of the BBC2008 model prove to be in very good agreement with the experimental results. The flexibility and the accuracy of the model together with the identification and integration procedure guarantee the applicability of the BBC2008 yield criterion in industrial applications.

The paper focuses on a reliable identification of the hardening curve of sheet metal after the onset of necking by means of a purposely developed shear test. The main comparative advantage of the test is its simple implementation needing no further development of the experimental equipment used in standard tensile testing. In addition, in order to achieve more flexible hardening curve description from those given by the classical analytical hardening laws, the paper presents the hardening curve approximation based on the cubic spline technique. The identified hardening curves are validated and compared to the analytically extrapolated ones obtained by a continuation of classical hardening laws for large plastic strain. It is shown, that due to large difference in response the analytical extrapolation is seriously disputable.

Strain recovery after removal of forming loads, commonly defined as springback, is of great concern in sheet metal forming, in particular with regard to proper prediction of the final shape of the part. To control the problem a lot of work has been done, either by minimizing the springback on the material side or by increasing the estimation precision in corresponding process simulations. Unfortunately, by currently available software springback still cannot be adequately predicted, because most analyses of springback are using linear, isotropic and constant Young’s modulus and Poisson’s ratio. But, as it was measured and reported, none of it is true. The aim of this work is to propose an upgraded mechanical model which takes evolution of damage and related orthotropic stiffness degradation into account. Damage is considered by inclusion of ellipsoidal cavities, and their influence on the stiffness degradation is taken in accordance with the Mori‐Tanaka theory, adopting the GTN model for plastic flow. Wit...

Strain recovery after removal of forming loads, commonly defined as springback, is of great concern in sheet metal forming, in particular with regard to proper prediction of the final shape of the part. To control the problem a lot of work has been done, either by minimizing the springback on the material side or by increasing the estimation precision in corresponding process simulations. Unfortunately, by currently available software springback still cannot be adequately predicted, because most analyses of springback are using linear, isotropic and constant Young’s modulus and Poisson’s ratio. But, as it was measured and reported, none of it is true. The aim of this work is to propose an upgraded mechanical model which takes evolution of damage and related orthotropic stiffness degradation into account. Damage is considered by inclusion of ellipsoidal cavities, and their influence on the stiffness degradation is taken in accordance with the Mori‐Tanaka theory, adopting the GTN model for plastic flow. In order to improve the numerical springback prediction, two major things are important: first, the correct evaluation of the stress‐strain state at the end of the forming process, and second, correctness of the elastic properties used in the elastic relaxation analysis. Since in modelling of the forming process we adopt a damage constitutive model with orthotropic stiffness degradation considered, a corresponding damage parameters identification upon specific experimental tests data must be performed first, independently of the metal forming modelling. An improved identification of material parameters, which simultaneously considers tensile test results with different type of specimens and using neural network, is proposed. With regard to the case in which damage in material is neglected it is shown in the article how the springback of a formed part differs, when we take orthotropic damage evolution into consideration.

When a metallic workpiece is subjected to heat treatment the crucial role in obtaining the final mechanical properties plays, apart from that of the initial chemical composition, the thermal history, ie. heating and cooling. Understanding of mechanisms and identification of parameters influencing the phase transformation kinetics is therefore of major importance for designing a heat treatment technology. The considered problem is characterized by extreme complexity and its investigation demands an interdisciplinary knowledge. Although a vast empirical knowledge about the problem, which is rather complex, has been gained, experience proves that it is often not sufficient. It seems that mathematical modelling, when based on experimentally verified models and adequately numerically modelled, could enable through a computer simulation an efficient design of more complex problems in a very objective and reliable way.