Many engineering design optimization problems involve multiple conflicting objectives, which today often are obtained by computational expensive finite element simulations. Evolutionary multi-objective optimization (EMO) methods based on surrogate modeling is one approach of solving this class of problems. In this paper, multi-objective optimization of a disc brake system to a heavy truck by using EMO and radial basis function networks (RBFN) is presented. Three conflicting objectives are considered. These are: 1) minimizing the maximum temperature of the disc brake, 2) maximizing the brake energy of the system and 3) minimizing the mass of the back plate of the brake pad. An iterative Latin hypercube sampling method is used to construct the design of experiments (DoE) for the design variables. Next, thermo-mechanical finite element analysis of the disc brake, including frictional heating between the pad and the disc, is performed in order to determine the values of the first two objectives for the DoE. Surrogate models for the maximum temperature and the brake energy are created using RBFN with polynomial biases. Different radial basis functions are compared using statistical errors and cross validation errors (PRESS) to evaluate the accuracy of the surrogate models and to select the most accurate radial basis function. The multi-objective optimization problem is then solved by employing EMO using the strength Pareto evolutionary algorithm (SPEA2). Finally, the Pareto fronts generated by the proposed methodology are presented and discussed.
In this paper, an approach to generate surrogate modelsconstructed by radial basis function networks (RBFN) with a prioribias is presented. RBFN as a weighted combination of radialbasis functions only, might become singular and no interpolationis found. The standard approach to avoid this is to add a polynomialbias, where the bias is defined by imposing orthogonalityconditions between the weights of the radial basis functionsand the polynomial basis functions. Here, in the proposed a prioriapproach, the regression coefficients of the polynomial biasare simply calculated by using the normal equation without anyneed of the extra orthogonality prerequisite. In addition to thesimplicity of this approach, the method has also proven to predictthe actual functions more accurately compared to the RBFNwith a posteriori bias. Several test functions, including Rosenbrock,Branin-Hoo, Goldstein-Price functions and two mathematicalfunctions (one large scale), are used to evaluate the performanceof the proposed method by conducting a comparisonstudy and error analysis between the RBFN with a priori and aposteriori known biases. Furthermore, the aforementioned approachesare applied to an engineering design problem, that ismodeling of the material properties of a three phase sphericalgraphite iron (SGI) . The corresponding surrogate models arepresented and compared
In order to obtain a robust performance, the established approach when using radial basis function networks (RBF) as metamodels is to add a posteriori bias which is defined by extra orthogonality constraints. We mean that this is not needed, instead the bias can simply be set a priori by using the normal equation, i.e. the bias becomes the corresponding regression model. In this paper we demonstrate that the performance of our suggested approach with a priori bias is in general as good as, or even for many test examples better than, the performance of RBF with a posteriori bias. Using our approach, it is clear that the global response is modelled with the bias and that the details are captured with radial basis functions. The accuracy of the two approaches are investigated by using multiple test functions with different degrees of dimensionality. Furthermore, several modeling criteria, such as the type of radial basis functions used in the RBFs, dimension of the test functions, sampling techniques and size of samples, are considered to study their affect on the performance of the approaches. The power of RBF with a priori bias for surrogate based design optimization is also demonstrated by solving an established engineering benchmark of a welded beam and another benchmark for different sampling sets generated by successive screening, random, Latin hypercube and Hammersley sampling, respectively. The results obtained by evaluation of the performance metrics, the modeling criteria and the presented optimal solutions, demonstrate promising potentials of our RBF with a priori bias, in addition to the simplicity and straight-forward use of the approach.
The automotive industry has shown a growing interest in tube hydroforming during the past years. The advantages of hydroforming (less thinning, a more efficient manufacturing process etc.) can, for instance, be combined with the high strength of extra high strength steels, which are usually less formable, to produce structural automotive components which exhibit lower weight and improved service performance. Design and production of tubular components require knowledge about tube material behaviour and tribological effects during hydroforming and how the hydroforming operation itself should be controlled. These issues are studied analytically in the present paper. Hydroforming consists of free forming and calibration. Only the so-called free forming is treated here. The analytical models constructed in this paper are used to show what the limits are during the free forming, how different material and process parameters influence the loading path and the forming result, and what an experimental investigation into hydroforming should focus on. The present study was a part of a larger investigation, in which finite-element simulations and experiments were also conducted. The results of these simulations and experiments will be accounted for in coming papers.
The springback of double curved autobody panels is studied theoretically and experimentally. Both steel and aluminum sheets are included in this investigation. The obtained results show that the springback is decreased with increasing binder force, increasing curvature, increasing sheet thickness and decreasing yield strength. This paper comprises also a discussion on the plastic strains and their influence on the springback.
Side Member Left and Side Member Right, which go from bumper to bumper, were at the focus in the present study. These side members were produced using straight round (hollow with a circular cross-section) extruded aluminium profiles as tube material. The tubes were bent and hydroformed. Rotary-draw bending yielded the best result. A spread within 8 mm after bending was found to be acceptable provided that the bent tube was hydroformed with a high maximum internal pressure (1300 bars in this study). Pressure-assisted tool (hydroforming tool) closure should be preferred. Such a tool closure prevents formation of buckles, which may be difficult to straighten out completely during hydroforming. Planeness and parallelity of the press tables and adapters play a significant role, as far as the spread and inplaneness of hydroformed components are concerned. The hydroforming tool must be matched in the press that actually will be used. Proper evacuation (of particularly air) is essential, especially in long hydroforming tools. All cross sections must be deformed at least 2% (average perimeter enlargement), if the hydroformed components are to exhibit a reasonable spread. The critical (fracture) cross-sections predicted by finite-element simulation correspond to those found in practical tests. However, the finite-element simulation was not able to predict formation of wrinkles at the tube ends caused by excessively large strokes. Such wrinkles were obtained in practice.
This work addresses the topology optimization approach to design robust compliant mechanisms with respect to uncertainties in the output stiffness, when compared to the traditional deterministic approach. To this end, two formulations are proposed: probabilistic and nonprobabilistic. The probabilistic formulation minimizes a joint objective function of expected output displacement plus a measure of its standard deviations, for given statistical distribution of the output stiffness. The nonprobabilistic formulation is written as minimization of a joint function of the median of output displacements, plus the width of the intervals that contains the extreme values of the output displacements, for a given interval of output stiffness. The Monte Carlo simulation method is used to evaluate expected values and standard deviations of output displacements in the probabilistic formulation and to assess results obtained with the deterministic approach. It is shown that both formulations lead to designs where output displacements are less sensitive to variations of output stiffness when compared to the traditional deterministic approach. Furthermore, as an additional benefit, it is observed that large variations of output stiffness can hinder the appearance of one-node connected hinges, usually found in the deterministic design of compliant mechanisms.
This paper presents two algorithms for solving the discrete, quasi-static, small-displacement, linear elastic, contact problem with Coulomb friction. The algorithms are adoptions of a Newton method for solving B-differentiable equations and an interior point method for solving smooth, constrained equations. For the application of the former method, the contact problem is formulated as a system of B-differentiable equations involving the projection operator onto sets with simple structure; for the application of the latter method, the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non-negativity of variables treated as constraints. The two algorithms are numerically tested for two-dimensional problems containing up to 100 contact nodes and up to 100 time increments. Results show that at the present stage of development, the Newton method is superior both in robustness and speed. Additional comparison is made with a commercial finite element code.
Topology optimization using stress constraints and considering uncertainties is aserious challenge, since a reliability problem has to be solved for each stress con-straint, for each element in the mesh. In this paper, an alternative way of solvingthis problem is used, where uncertainty quantification is performed through thefirst-order perturbation approach, with proper validation by Monte Carlo simulation.Uncertainties are considered in the loading magnitude and direction. The mini-mum volume problem subjected to local stress constraints is formulated as a robustproblem, where the stress constraints are written as a weighted average betweentheir expected value and standard deviation. The augmented Lagrangian method isused for handling the large set of local stress constraints, whereas a gradient-basedalgorithm is used for handling the bounding constraints. It is shown that even in thepresence of small uncertainties in loading direction, different topologies are obtainedwhen compared to a deterministic approach. The effect of correlation between uncer-tainties in loading magnitude and direction on optimal topologies is also studied,where the main observed result is loss of symmetry in optimal topologies.
Topology optimization of continuum structures is a challenging problem to solve, when stress constraints are considered for every finite element in the mesh. Difficulties are compounding in the reliability-based formulation, since a probabilistic problem needs to be solved for each stress constraint. This paper proposes a methodology to solve reliability-based topology optimization problems of continuum domains with stress constraints and uncertainties in magnitude of applied loads considering the whole set of local stress constrains, without using aggregation techniques. Probabilistic constraints are handled via a first-order approach, where the principle of superposition is used to alleviate the computational burden associated with inner optimization problems. Augmented Lagrangian method is used to solve the outer problem, where all stress constraints are included in the augmented Lagrangian function; hence sensitivity analysis may be performed only for the augmented Lagrangian function, instead of for each stress constraint. Two example problems are addressed, for which crisp black and white topologies are obtained. The proposed methodology is shown to be accurate by checking reliability indices of final topologies with Monte Carlo Simulation.
This work addresses the use of the topology optimization approach to the design of robust continuum structures under the hypothesis of uncertainties with known second order statistics. To this end, the second-order perturbation approach is used to model the response of the structure and the midpoint discretization technique is used to discretize the random field. The objective function is a weighted sum of the expected compliance and its standard deviation. The optimization problem is solved using a traditional optimality criteria method. It is shown that the correlation length plays an important role in the obtained topology and statistical moments when only the minimization of the standard deviation is considered, resulting in more and thinner reinforcements as the correlation length decreases. It is also shown that the minimization of the expected value is close to the minimization of the deterministic compliance for small variations of Young's modulus.
This work addresses the use of the topology optimization approach to the design of continuum structures with failure constraints under the hypothesis of uncertainties in the spatial distribution of Young’s modulus. To this end, the first order perturbation approach is used to model the response of the structure and the midpoint discretization technique is used to represent the random field. The objective is the minimization of the amount of material used in the design, subjected to local stress constraints under uncertainties. The probability of failure is bounded by the one-sided Chebychev inequality, since the exact probability distribution function of the stress constraints is not known in advance. The effective probability of failure of the obtained optimal designs is validated with the use of the Monte Carlo Simulation approach, indicating that the probability of failures of the topologies obtained with the stochastic approach is within the bounds provided by the one-sided Chebychev inequality. The optimization problem is solved by means of the augmented Lagrangian method, in order to address the large number of constraints associated to this kind of formulation. It is shown that the correlation length and the number of standard deviations considered in the formulation play an important role in both the obtained topology and effective probability of failure.
This work reports the evaluation of a tactile sensor for the potential of using it to measure the compliance of folding cartons. A tactile sensor would make it possible to measure the mechanical behavior locally around the contact point, in contrast to existing methods that measure the global mechanical behavior of the carton. Research on the haptic sense has shown that the local mechanical behavior is more important than the global behavior when humans assess compliance of objects. It is shown that the response of the tactile sensor correlates strongly with the bending stiffness of the board, but also with geometric features. A method for reducing the 22-dimensional output of the sensor to single meaningful feature using linear discriminant analysis is proposed and tested. The results show that the sensor is a good candidate for a method that incorporates both cutaneous and kinaesthetic information in the measure of carton compliance.
Carton board packages subjected to concentrated loads near the edges are damaged in a characteristic way. This paper reports an attempt to simulate the damage process in the lab and Package Collapse Loads measured for this load case.
Packages were compressed by a concentrated load. The position of load application was varied along a line parallel to a crease and the package was rotated in order to test the influence of the height of the load carrying panel. Force and displacement were recorded and the damage evolution during the test was studied. The damage produced was examined using x-ray tomography. The nature of damage at different stages of damage evolution was studied.
Both the visual appearance of the damage and the force-displacement curve were similar in all tests. The Package Collapse Load has little dependence on where along a line parallel to a crease of the package the point load is applied. Damage started developing at the crease and a yield line perpendicular to the crease and parallel to the direction of the load developed. When the displacement increased further, a parabolic yield line, symmetric around the previous one, developed. The start of the damage development was associated with at peak in the force-displacement curve. Stiffness was more geometry dependent than strength. On macro scale, the visual appearance of the damage due to concentrated loads shows no significant dependence on geometry.
The importance of sensory information in product purchasing decisions has gained increasing attention in recent years. Tactile properties of packaging are usually measured with the help of trained evaluators. An objective, fast and repeatable method that describes the mechanical interaction and does not rely on a panel would have many benefits. We propose and evaluate such a method for measuring the mechanical interaction between a deformable finger-like shaped sensor and a package. Evaluation of the method shows good repeatability, the variability in the measurement result is within a few percent in most cases. The method captures indentation differences at contact between sensor and package due to measurement position and package design.
In this work residual stresses in a stress lattice are studied. The residual stresses are both measured and simulated. The stress lattice is casted of low alloyed grey cast iron. In fact, nine similar lattices are casted and measured. The geometry of the lattice consists of three sections in parallel. The diameter of the two outer sections are thinner than the section in the middle. When the stress lattice cools down, this difference in geometry yields that the outer sections start to solidify and contract before the section in the middle. Finally, an equilibrium state, with tensile stresses in the middle and compressive stresses in the outer sections, is reached. The thermo-mechanical simulation of the experiments is performed by using Abaqus. The thermo-mechanical solidification is assumed to be uncoupled. First a thermal analysis, where the lattice is cooled down to room temperature, is performed. Latent heat is included in the analysis by letting the fraction of solid be a linear function of the temperature in the mushy zone. After the thermal analysis a quasi-static mechanical analysis is performed where the temperature history is considered to be the external force. A rate independent J2-plasticity model with isotropic hardening is considered, where the material data depend on the temperature. Tensile tests are performed at room temperature, 200°C, 400°C, 600°C and 800°C in order to evaluate the Young´s modulus, the yield strength and the hardening accurate. In addition, the thermal expansion coefficient is evaluated for temperatures between room temperature and 1000°C. The state of residual stresses is measured by cutting the mid section or the outer section. The corresponding elastic spring-back reveals the state of residual stresses. The measured stresses are compared to the numerical simulations. The simulations show good agreement with the results from the experiments.
In this paper, an optimization routine for a thermomechanical problem is presented. The optimization routine is based on the successive response surface methodology where the panning and zooming technique presented by Stander and Craig has been implemented and improved. The optimization routine has been applied to an optimization problem of a three-dimensional beam that undergoes a solidification process. The material in the beam is assumed to be low-alloyed gray iron. The thermomechanical solidification analysis is uncoupled where, first, a thermal analysis is performed to determine the thermal history. This thermal history is then used to calculate the residual stresses in the beam. The residual stresses are solved by using classical J(2)-plasticity with temperature-dependent material properties. The residual stresses from solidification are then carried on to the structural analysis where a mechanical load is applied. These are all linked together via scripts, and the commercial FE software Abaqus is used as the FE solver. The obtained maximum von Mises stress and mass information for every set of parameters are then exported to Matlab where general quadratic response surfaces are fitted by a least square method. Taken together, these response surfaces define a minimum of weight problem, which is solved by using sequential linear programming. To minimize the number of evaluations needed, the parameters are chosen to be D-optimally selected. The numerical results show that the residual stresses from solidification might influence the optimal shape significantly. The residual stress results have been compared with those obtained from casting simulation softwares, and the results are similar. The optimization has been compared with a commercial optimization software and shows very promising results.
In this work a general method for structural optimization of nonlinear structures is implemented using ABAQUS and Matlab. The method utilizes the response surface methodology with polynomial surfaces and nonlinear programming. In such manner a method that is applicable for a large number of different classes of nonlinear problems is obtained. For instance plasticity problems, thermomechanical problems and contact problems can be optimized using this strategy. In this paper, the method is utilized to minimize weight of castings by including residual stresses from solidification. This is performed by first determine the residual stresses by a thermomechanical analysis of a metal structure that is cooled down from a temperature above liquidus temperature down to room temperature. These residual stresses are then included when the problem of minimum of weight is formulated. The shape of the structure will of course affect the residual stress distribution during the optimization and the optimal shape will be different from the one obtained when residual stresses are not included in the analysis. The method is implemented by using a Python script and m-files. In such way a parameterized model can easily be treated in ABAQUS and Matlab during the optimization process. The parameterized geometry, loads, boundary conditions and mesh are first generated by the ABAQUS/CAE module. The nonlinear models are then solved using ABAQUS/Standard. A set of solutions are generated by solving the model for a pre-defined set of parameters. In order to minimize the number of simulations and still achieve good surface approximations these parameters are taken to be D-optimal. The sets of solutions and parameters are in turn exported to Matlab where general quadratic response surfaces are fitted by the least square method. By utilizing these surfaces the problem of minimum of weight subjected to constraints on stresses is formulated. Finally, the nonlinear optimization problem is solved by sequential linear programming where the linear part is solved using Matlab.
In this work a general method for structural optimization of nonlinear structures is implemented using FE-analysis. The method utilizes the response surface methodology with polynomial surfaces and nonlinear programming. In such manner a method that is applicable for a large number of different classes of nonlinear problems is obtained. In this paper, the method is utilized to minimize weight of castings by including residual stresses from solidification. This is performed by first determine the residual stresses by a thermomechanical analysis of a metal structure that is cooled from a temperature above liquidus temperature down to room temperature. The thermomechanical analysis is uncoupled where the temperature distribution within the casting as a function of time is determined first and is later on used for residual stress calculations. These residual stresses are then included when the mechanical load is applied to the structure and the problem of minimum of weight is formulated. The structure shown in this paper is an example of a two dimensional geometry. The shape of the structures will of course affect the residual stress distribution during the optimization. The nonlinear models are then solved using ABAQUS/Standard. A set of solutions are generated by solving the model for a pre-defined set of parameters. In order to minimize the number of simulations and still achieve good surface approximations these parameters are taken to be D-optimal. The sets of solutions and parameters are in turn exported to Matlab where general quadratic response surfaces are fitted by the least square method. By utilizing these surfaces the problem of minimum of weight subjected to constraints on stresses is formulated. Finally, the nonlinear optimization problem is solved by sequential linear programming.
The mechanical behaviour of adhesive joints is critical for the performance of adhesively joined carton board packages. In this work, finite element analyses of hot melt adhesive (HMA) joints in carton board is conducted and compared to experimental results obtained using a Y-peel testing device. The aim of the present study is to analyse the behaviour of adhesive joints tested in the Y-peel testing device using a layered carton board model.
The carton board is modelled as a layered structure where the layers are assumed to obey Hill's orthotropic elastic-plastic model, and the interfaces are modelled using a softening orthotropic damage model. The HMA is modelled as isotropic linear elastic, and the influence from a varying elastic modulus of the HMA is explored. It is found that the pre-peak behaviour of the Y-peel force-elongation curves is reasonably well captured by the FE simulations, although the initial stiffness is somewhat too high. Also, the pre-peak behaviour is practically insensitive to changes of the elastic modulus of the HMA.
The deformation and delamination pattern obtained in the simulations was compared with microscope pictures taken during the corresponding Y-peel experiments, and it is shown that they conform to the observed behaviour during Y-peel testing at comparable loading levels. However, the delamination opening is somewhat underestimated by the model.
In this paper, we propose a stabilized augmented Lagrange multiplier method for the finite element solution of small deformation elastic contact problems. We limit ourselves to friction–free contact with a rigid obstacle, but the formulation is readily extendable to more complex situations.
In this work a hybrid method of a genetic algorithm and sequential linear programming is suggested to obtain a D-optimal design of experiments. Regular as well as non-regular design spaces are considered. A D-optimal design of experiments maximizes the determinant of the information matrix, which appears in the normal equation. It is known that D-optimal design of experiments sometimes include duplicate design points. This is, of course, not preferable since duplicates do not add any new information to the response surface approximation and the computational effort is therefore wasted. In this work a Bayesian modification, where higher order terms are added to the response surface approximation, is used in case of duplicates in the design of experiments. In such manner, the draw-back with duplicates might be eliminated. The D-optimal problem, which is obtained by using the Bayesian modification, is then solved by a hybrid method. A hybrid method of a genetic algorithm that generates a starting point for sequential linear programming is developed. The genetic algorithm performs genetic operators such as cross-over and mutation on a binary version of the design of experiments, while the real valued version is used to evaluate the fitness. Next, by taking the gradient of the objective, a LP-problem is formulated which is solved by an interior point method that is available in Matlab. This is repeated in a sequence until convergence is reached. The hybrid method is tested for four numerical examples. Results from the numerical examples show a very robust convergence to a global optimum. Furthermore, the results show that the problem with duplicates is eliminated by using the Bayesian modification.
In this work the robustness of residual stresses in finite element simulations with respect to deviations in mechanical parameters in castings is evaluated. Young's modulus, the thermal expansion coefficient and the hardening are the studied parameters. A 2D finite element model of a stress lattice is used. The robustness is evaluated by comparing purely finite element based Monte Carlo simulations and Monte Carlo simulations based on linear and quadratic response surfaces. Young's modulus, the thermal expansion coefficient and the hardening are assumed to be normal distributed with a standard deviation that is 10% of their nominal value at different temperatures. In this work an improved process window is also suggested to show the robustness graphically. By using this window it is concluded that least robustness is obtained for high hardening values in combination to deviations in Young's modulus and the thermal expansion coefficient. It is also concluded that quadratic response surface based Monte Carlo simulations substitute finite element based Monte Carlo simulations satisfactory. Furthermore, the standard deviation of the responses are evaluated analytically by using the Gauss formula, and are compared to results from Monte Carlo simulations. The analytical solutions are accurate as long as the Gauss formula is not utilized close to a stationary point.
In the past stamping dies have in principle been designed by rules of thumb and intuition. As the sheet metals in the vehicle industry have got increased mechanical properties in recent years the demands on the stamping dies have increased. For instance increase in stiffness is desirable in order to better control spring-back. The most simple way to satisfy this new demand would be to make the stamping dies even more heavy in order to be able to handle the new sheet metals. Since there are restrictions of the weight of the stamping dies in the stamping machines and since the overhead cranes usually have reached the limit of what they can handle, this is not a desirable solution. Another approach, in order to increase the stiffness without increasing the weight is to use topology optimization. Recently in a master thesis at Volvo Car Corporation a conceptual design of a stamping die has been done by topology optimization. In that work no consideration is taken to the fact that the stamping die is casted. Casting implies that residual stresses possibly are produced during the solidification and cooling process. The residual stresses might affect the fatigue life and the risk of failure of the stamping die.
In this work the residual stress state after casting is analyzed for the original stamping die as well as the optimized stamping die from the master thesis discussed above. The analyses are performed using an uncoupled approach, where one thermal analysis is followed by a quasi-static elasto-plastic analysis. The thermal analysis simulates the solidification and cooling during the casting process, while the quasi-static elasto-plastic analysis uses the temperature history, obtained from the thermal analysis, in order to build up residual stresses. The thermal analysis includes the release of latent heat. Furthermore, the material properties included in the heat equation (density, conductivity, specific heat) are given as temperature dependent properties for the mould as well as the casting. In the quasi-static elasto-plastic analysis the plasticity is described by the von Mises yield surface in combination with isotropic hardening and the mechanical properties (thermal expansion coefficient, Young's modulus, yield stress, hardening parameter, Poisson's ratio) are given as temperature dependent properties. The simulations show high levels of residual stresses.
In this paper a polynomial regression model where the constituents of are of arbitrary order is proposed. A genetic algorithm is used to find the optimal terms to be included in the so-called optimal polynomial regression model . The objective for the genetic algorithm is to minimize the sum of squared errors of the predicted responses. In practice the genetic algorithm generates an optimal set of exponents of the design variables in a polynomial regression model. Several example problems are presented to show the performance and accuracy of the optimal polynomial regression model. Results show a greatly improved performance for optimal polynomial regression models compared to traditional regression models.
In mobile robot applications, some sensors such as open-path gas detectors or laser rangefinders need to be aimed at specific targets in order to get the desired measurements. To do this in a fast and elegant manner, we present a spherical parallel manipulator with three degrees of freedom. Compared to typical serial manipulators, it offers superior dynamics and structural stiffness, which are important parameters for this type of task. We present the mechanical design and derive kinematic equations both to compute set-points for the desired orientation and to estimate the current state of the system. A PID controller is used to generate control signals.
In robotic applications, it is often necessary to orient a sensor quickly. Spherical parallel manipulators (SPM) are well suited for this purpose since they offer superior dynamics and structural stiffness as compared to serial manipulators. To control them, however, the kinematic equations have to be known. In this paper, a SPM with three degrees of freedom and the kinematic equations describing its mechanical properties are presented.
In this paper finite element approaches for fretting fatigue are proposed on the basis of a non-local model of continuum damage coupled to friction and wear. The model is formulated in the frame-work of a standard material. In a previous paper this was done in the spirit of Maugin, where an extra entropy flux is introduced in the second law in order to include the gradient of the internal variable in a proper manner. In this paper we follow instead the ideas of Frémond and others, where this extra entropy flux is no longer needed, but instead new non-classical balance laws associated to damage, friction and wear, respectively, are derived from the principle of virtual power. The standard material is then defined as usual by state laws based on free energies and complementary laws based on dissipation potentials. In particular, we pick free energies and dissipation potentials that correspond to a non-local continuum damage model coupled to friction and wear. In addition, the boundary conditions at the contact interface creates a coupling between damage and wear. This is a key feature of our model, which makes it very useful in studies of fretting fatigue. By starting from a variational formulation of the governing equations, two different finite element algorithms are implemented. Both algorithms are based on a Newton method for semi-smooth equations. In the first algorithm the Newton method is applied to the entire system of equations, while in the second algorithm the system of equations is split into two different parts such that an elastic wear problem is solved for fixed damage followed by the solution of the damage evolution problem for the updated displacements and contact forces in an iterative process. The latter algorithm can be viewed as a Gauss-Seidel scheme. The numerical performance of the algorithms is investigated for three twodimensional examples of increasing complexity. Based on the numerical solutions, the behavior of the model is also discussed. For instance, it is shown numerically how the initiation of damage depends on the contact geometry, the coefficient of friction and the evolution of wear.
The present paper presents a model of damage coupled to wear. The damage model is based on a continuum model including the gradient of the damage variable. Such a model is non-local in the sense that the evolution of damage is governed by a boundary-value problem instead of a local evolution law. Thereby, the well-known mesh-dependency observed for local damage models is removed. Another feature is that the boundary conditions can be used to introduce couplings between bulk damage and processes at the boundary. In this work such a coupling is suggested between bulk damage and wear at the contact interface. The model is regarded as a first attempt to formulate a continuum damage model for studying crack initiation in fretting fatigue.
The model is given within a thermodynamic framework, where it is assured that the principles of thermodynamics are satisfied. Furthermore, two variational formulations of the full. initial boundary value problem, serving as starting points for finite element discretization, are presented. Finally, preliminary numerical results for a simple one-dimensional example are presented and discussed. It is qualitatively shown how the evolution of damage may influence the wear behaviour and how damage may be initiated by the wear process.
This contribution concerns the numerical treatment of discrete thermoelastic wear problems. Two different approaches, both utilizing a non-smooth Newton method as non-linear equation solver, are outlined and compared. Furthermore, a numerical example shows how the predicted wear gap is influenced by the bulk properties of the contacting bodies.
In the present paper three algorithms are applied to a finite element model of two thermoelastic bodies in frictional wearing contact. All three algorithms utilize a modification of a Newton method for B-differentiable equations as non-linear equation solver. In the first algorithm the fully-coupled system of thermomechanical equations is solved directly using the modified method, while in the other two algorithms the equation system is decoupled in one mechanical part and another thermal part which are solved using an iterative strategy of Gauss–Seidel type. The two iterative algorithms differ in which order the parts are solved. The numerical performance of the algorithms are investigated for two two-dimensional examples. Based on these numerical results, the behaviour of the model is also discussed. It is found that the iterative approach where the thermal subproblem is solved first is slightly more efficient for both examples. Furthermore, it is shown numerically how the predicted wear gap is influenced by the bulk properties of the contacting bodies, in particular how it is influenced by thermal dilatation.
In this work, generative design optimization and characterization of triple periodic lattice structures in AlSi10Mg are considered. Structures with Gyroid, Schwarz-D and G-prime lattices are designed optimally by utilizing a generative design optimization approach. The approach is based on topology optimization, support vector machines (SVM), radial basis function networks (RBFN), morphing operations, design of experiments and metamodels. Firstly, topology optimization solutions are generated which are represented using SVM, secondly, sizing solutions obtained by setting the SIMP parameter equal to one are represented with RBFN. Thirdly, graded lattice structures using the RBFN are morphed together with the SVM to final conceptual designs. Fourthly, design of experiments of the conceptual designs are performed using non-linear finite element analyses (FEA) and, finally, metamodel-based design optimization is conducted using convex combinations of Kriging, RBFN, polynomial chaos expansion and support vector regression models. In order to validate the optimal designs, new tensile test specimens that include the periodic lattice structures are suggested. The specimens with all three lattices are manufactured in AlSi10Mg using direct metal laser sintering with an EOS M290 machine. Tensile tests of these specimens are then performed and validated using nonlinear FEA. The test specimens are also characterized with respect to geometry and defects by means of computed tomography, optical microscopy and scanning electron microscopy. The study demonstrates the high potential of using the proposed generative design optimization approach with triple periodic lattice structures for producing robust lightweight designs using additive manufacturing. In order to demonstrate the industrial relevance the established GE engine bracket is studied in the paper and discussed at the conference.
In manufacturing industry there is a high demand for on line quality control to minimize the risk of incorrectly produced objects. Conventional contact measurement methods are usually slow and invasive, meaning that they cannot be used for soft materials and for complex shapes without influencing thecontrolled parts. In contrast, interferometry and digital holography in combination with computers become faster, more reliable and highly accurate as an alternative non-contact technique for industrial shape evaluation. In digital holography, access to the complex wave field and the possibility tonumerically reconstruct holograms in different planes introduce a new degree of flexibility to optical metrology. With digital holography high resolution and precise three dimensional (3D) images of the manufactured parts can be generated. This technique can also be used to capture data in a single exposure,which is important when doing measurements in a disturbed environment.The aim of this thesis is to perform online process control of free-form manufactured objects by measuring the shape and compare it to the CAD-model. To do this, a new technique to measure surface gradients and shape based on single-shot dual wavelength digital holography and image correlation of speckle displacements is demonstrated. Based on an analytical relation between phase gradients and speckle displacements it is shown that an object is retrieved uniquely to shape and position without the unwrapping problems that usually appear in dual wavelength holography. The method is firstdemonstrated using continues wave laser light from two temperature controlled laser diodes operating at 640 nm. Further a specially designed dual core diode pumped fiber laser that produces pulsed light with wavelengths close to 1030 nm is used. One significant problem when using the dual wavelength single-shot approach is that phase ambiguities are built in to the system that needs to be corrected. An automatic calibration scheme is therefore required. The intrinsic flexibility of digital holography gives a possibility to compensate these aberrations and to remove errors, fully numerically without mechanical movements. In this thesis I present a calibration method which allows single-shot online shape evaluation in a disturbed environment. It is shown that phase maps and speckle displacements can be recovered free of chromatic aberrations. This is the first time that a single-shot dual wavelength calibration is reported by defining a criteria to make an automatic procedure.By the results of the presented work, it is experimentally verified that the single-shot dual wavelength digital holography and numerically generated speckle images can be used together with digital speckle correlation to retrieve and evaluate the object shape. The proposed method is also robust to large phasegradients and large movements within the intensity patterns. The advantage of the approach is that, using speckle displacements, the shape measurement can be done even though the synthetic wavelength is out of the dynamic range of the height variation of the object.
In manufacturing industry there is a high demand for on line quality control to minimize the risk of incorrectly produced objects. Conventional contact measurement methods are usually slow and invasive, meaning that they cannot be used for soft materials and for complex shapes without influencing the controlled parts. In contrast, interferometry and digital holography in combination with computers become faster, more reliable and highly accurateas an alternative non-contact technique for industrial shape evaluation. For example in digital holography, access to the complex wave field and the possibility to numerically reconstruct holograms in different planes introduce a new degree of flexibility to optical metrology. With digital holography high resolution and precise three dimensional (3D) images of the manufactured parts can be generated. This technique can also be used to capture data in asingle exposure, which is important when doing measurements in a disturbed environment. The aim of this thesis is devoted to the theoretical and experimental development of shape and deformation measurements. To perform online process control of free-form manufactured objects, the measured shape is compared with the CAD-model to obtain deviations. To do this, a new technique to measure surface gradients and shape based onsingle-shot multiplexed dual-wavelength digital holography and image correlation of speckle displacements is demonstrated. Based on an analytical relation between phase gradients and speckle displacements it is shown that an object is retrieved uniquely to shape, position and deformation without the unwrapping problems that usually appear in dual-wavelength holography. The method is first demonstrated using continues-wave laser light from twotemperature controlled laser diodes operating at 640 nm. Then a specially designed dual core diode pumped fiber laser that produces pulsed light with wavelengths close to 1030 nm is used. In addition, a Nd:YAG laser with the wavelength of 532 nm is used for 3D deformation measurements. One significant problem when using the dual-wavelength single-shot approach is that phase ambiguities are built in to the system that needs to be corrected. An automatic calibration scheme is therefore required. The intrinsic flexibility of digital holography gives a possibility to compensate these aberrations and to remove errors, fully numerically without mechanical movements. In this thesis I present a calibration method which allows multiplexed singleshotonline shape evaluation in a disturbed environment. It is shown that phase maps and speckle displacements can be recovered free of chromatic aberrations. This is the first time that a multiplexed single-shot dual-wavelength calibration is reported by defining a criteria tomake an automatic procedure. Further, Digital Speckle Photography (DSP) is used for the full field measurement of 3D deformations. In order to do 3D deformation measurement, usually multi-cameras and intricate set-up are required. In this thesis I demonstrate the use of only one single camera torecord four sets of speckle patterns recorded by illuminating the object from four different directions. In this manner, meanwhile 3D speckle displacement is calculated and used for the measurement of the 3D deformations, wrapping problems are also avoided. Further, the same scale of speckle images of the surface for all four images is guaranteed. Furthermore, a need for calibration of the 3D deformation measurement that occurs in the multi-camera methods,is removed. By the results of the presented work, it is experimentally verified that the multiplexed singleshot dual wavelength digital holography and numerically generated speckle images can be used together with digital speckle correlation to retrieve and evaluate the object shape. Usingmultidirectional illumination, the 3D deformation measurements can also be obtained. The proposed method is robust to large phase gradients and large movements within the intensity patterns. The advantage of the approach is that, using speckle displacements, shape and deformation measurements can be performed even though the synthetic wavelength is out of the dynamic range of the object deformation and/or height variation.
The present theoretical note shows how a naturalobjective function in stiffness optimization, including bothprescribed forces and non-zero prescribed displacements,is the equilibrium potential energy. It also shows how theresulting problem has a saddle point character that may beutilized when calculating sensitivities.
Stiffness topology optimization is usually based on a state problem of linear elasticity, and there seems to be little discussion on what is the limit for such a small rotation-displacement assumption. We show that even for gross rotations that are in all practical aspects small (<3 deg), topology optimization based on a large deformation theory might generate different design concepts compared to what is obtained when small displacement linear elasticity is used. Furthermore, in large rotations, the choice of stiffness objective (potential energy or compliance), can be crucial for the optimal design concept. The paper considers topology optimization of hyperelastic bodies subjected simultaneously to external forces and prescribed non-zero displacements. In that respect it generalizes a recent contribution of ours to large deformations, but we note that the objectives of potential energy and compliance are no longer equivalent in the non-linear case. We use seven different hyperelastic strain energy functions and find that the numerical performance of the Kirchhoff–St.Venant model is in general significantly worse than the performance of the other six models, which are all modifications of this classical law that are equivalent in the limit of infinitesimal strains, but do not contain the well-known collapse in compression. Numerical results are presented for two different problem settings.
In the present paper, a large rotational approach for dynamic contact problems with friction is proposed. The approach is used for modelling a spur gear pair with shafts and bearings. The model is obtained by superposing small displacement elasticity on rigid-body motions, and postulating tribological laws on the gear flanks. The finite element method is used to model the elastic properties of the gear pair. Shafts and bearings are represented by linear springs. The tribological laws of the contact interface are Signorini's contact law and Coulomb's law of friction. An important feature of the approach is that the difficulties of impacting mass nodes are avoided. The governing equations of the model are numerically treated by use of the augmented Lagrangian approach. In such manner the geometry of the gear flanks are well represented in the numerical simulations. It is possible to study accurately the consequences of different types of profile modifications as well as flank errors. In this work, the dynamic transmission error is studied. For instance, it turns out that the effect from profile modification is less significant for the transmission error when frictional effects are included.
The perception of mechanical rigidity when touching a package is important for purchasing decisions. This perception will depend both on the material and geometry of the product packaging, but also on the position where the package is grasped. Both kinaestethic (globally) and cutaneous cues (locally around the fingertip) play a role in the perception of compliance, but cutaneous cues are more important. We therefore use a tactile sensor to investigate the mechanical interaction between the tactile sensor and a cartonboard package; we study the changes depending on the measuring position and the material. Using linear discriminant analysis (LDA) on the measurement result we show that we can separate these two changes for separate analysis.
Carton board packages with different creases have been manufactured and loaded with point loads to see if there is a difference in point load behavior depending on crease. A carton board was creased in four different ways. The differences of the creases were judged according to appearance and residual moment. Three of the four creases showed divergent appearances. The re-sidual moment was as expected lowest for the hardest creased sample and vice versa. The packages with the different creases were loaded with a point load. The registered force - displacement curves were analyzed according to stiffness and collapse load. No statistically significant difference could be seen in the collapse load of the package or in the stiffness measured, indicating that with a standard point load measurement it was hard to distinguish a difference in packaging behavior due to creasing. In future work it might be interesting to further study the average stiffness, since a trend of higher average stiffness for lower matrix channel depth exists in the measurements, however not statistically ensured. One possibility to discern possibly existing differences in the resistance of packaging to point load due to crease differences could be to study the interaction in more detail.
In this paper an efficient approach to simulate thermal stresses due to frictional heating of disc brakes is presented. Inthe approach thermal and stress analysis are performed sequentially. The frictional heat analysis is based on the Eulerianmethod, which requires significantly low computational time as compared to the Lagrangian approach. Completethree-dimensional geometries of a disc and a pad are considered for the numerical simulations. The contact forcesare computed at each time step taking the thermal deformations of the disc into account. The nodal temperaturehistory is recorded at each time step and is used in sequentially coupled stress analysis, where a temperature dependentelasto-plastic material model is used to compute the stresses in a disc brake. The results show that during hard braking,high compressive stresses are generated on the disc surface in circumferential direction which cause plastic yielding. Butwhen the disc cools down, the compressive stresses transform to tensile stresses. Such thermoplastic stress history maycause cracks on disc surface after a few braking cycles. These results are in agreement with experimental observationsavailable in the literature.
In this paper, an efficient sequential approach for simulating thermal stresses in brake discs for repeated braking is presented. First, a frictional heat analysis is performed by using an Eulerian formulation of the disc. Then, by using the temperature history from the first step of the sequence, a plasticity analysis with temperature dependent material data is performed in order to determine the corresponding thermal stresses. Three-dimensional geometries of a disc and a pad to a heavy truck are considered in the numerical simulations. The contact forces are computed at each time step taking the thermal deformations of the disc and pad into account. In such manner, the frictional heat power distribution will also be updated in each time step, which in turn will influence the development of heat bands. The plasticity model is taken to be the von Mises yield criterion with linear kinematic hardening, where both the hardening and the yield limit are temperature dependent. The results show that during hard braking, high compressive stresses are generated on the disc surface in the circumferential direction which cause yielding. But when the disc cools down, these compressive stresses transform to tensile residual stresses. For repeated hard braking when this kind of stress history is repeated, we also show that stress cycles with high amplitudes are developed which might generate low cycle fatigue cracks after a few braking cycles.
In this paper frictional heating of a disc brake is simulated while taking wear into account. By performing thermomechanical finite element analysis, it is studied how the wear history will influence the development of hot bands. The frictional heat analysis is based on an Eulerian formulation of the disc, which requires significantly lower computational time as compared to a standard Lagrangian approach. A real disc-pad system to a heavy truck is considered, where complete three-dimensional geometries of the ventilated disc and pad are used in the simulations. A sequential approach is adopted, where the contact forces are computed at each time step taking the wear and thermal deformations of the mating parts into account. After each brake cycle, the wear profile of the pad is updated and used in subsequent analysis. The results show that when wear is considered, different distributions of the temperature on disc are obtained for each new brake cycle. After a few braking cycles two hot bands appear on the disc surface instead of only one. These results are in agreement with experimental observations.