ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar
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2019,2
Due to an increased need for hydro-electricity, water storage, and flood protection, it is assumed that a series of new dams will be built throughout the world. Comparing existing design methodologies for arch-type dams, model-based shape optimization can effectively reduce construction costs and leverage the properties of construction materials. To apply the means of shape optimization, suitable variables need to be chosen to formulate the objective function, which is the volume of the arch dam here. In order to increase the consistency with practical conditions, a great number of geometrical and behavioral constraints are included in the mathematical model. An optimization method, namely Genetic Algorithm is adopted which allows a global search.
Traditional optimization techniques are realized based on a deterministic approach, which means that the material properties and loading conditions are assumed to be fixed values. As a result, the real-world structures that are optimized by these approaches suffer from uncertainties that one needs to be aware of. Hence, in any optimization process for arch dams, it is nec- essary to find a methodology that is capable of considering the influences of uncertainties and generating a solution which is robust enough against the uncertainties.
The focus of this thesis is the formulation and the numerical method for the optimization of the arch dam under the uncertainties. The two main models, the probabilistic model, and non-probabilistic models are intro- duced and discussed. Classic procedures of probabilistic approaches un- der uncertainties, such as RDO (robust design optimization) and RBDO (reliability-based design optimization), are in general computationally ex- pensive and rely on estimates of the system’s response variance and fail- ure probabilities. Instead, the robust optimization (RO) method which is based on the non-probabilistic model, will not follow a full probabilistic approach but works with pre-defined confidence levels. This leads to a bi-level optimization program where the volume of the dam is optimized under the worst combination of the uncertain parameters. By this, robust and reliable designs are obtained and the result is independent of any as- sumptions on stochastic properties of the random variables in the model.
The optimization of an arch-type dam is realized here by a robust optimiza- tion method under load uncertainty, where hydraulic and thermal loads are considered. The load uncertainty is modeled as an ellipsoidal expression. Comparing with any traditional deterministic optimization (DO) method, which only concerns the minimum objective value and offers a solution candidate close to limit-states, the RO method provides a robust solution against uncertainties.
All the above mentioned methods are applied to the optimization of the arch dam to compare with the optimal design with DO methods. The re- sults are compared and analyzed to discuss the advantages and drawbacks of each method.
In order to reduce the computational cost, a ranking strategy and an ap- proximation model are further involved to do a preliminary screening. By means of these, the robust design can generate an improved arch dam structure which ensures both safety and serviceability during its lifetime.
2022,6
In this thesis, a new approach is developed for applications of shape optimization on the time harmonic wave propagation (Helmholtz equation) for acoustic problems. This approach is introduced for different dimensional problems: 2D, 3D axi-symmetric and fully 3D problems. The boundary element method (BEM) is coupled with the isogeometric analysis (IGA) forming the so-called (IGABEM) which speeds up meshing and gives higher accuracy in comparison with standard BEM. BEM is superior for handling unbounded domains by modeling only the inner boundaries and avoiding the truncation error, present in the finite element method (FEM) since BEM solutions satisfy the Sommerfeld radiation condition automatically. Moreover, BEM reduces the space dimension by one from a volumetric three-dimensional problem to a surface two-dimensional problem, or from a surface two-dimensional problem to a perimeter one-dimensional problem. Non-uniform rational B-splines basis functions (NURBS) are used in an isogeometric setting to describe both the CAD geometries and the physical fields.
IGABEM is coupled with one of the gradient-free optimization methods, the Particle Swarm Optimization (PSO) for structural shape optimization problems. PSO is a straightforward method since it does not require any sensitivity analysis but it has some trade-offs with regard to the computational cost. Coupling IGA with optimization problems enables the NURBS basis functions to represent the three models: shape design, analysis and optimization models, by a definition of a set of control points to be the control variables and the optimization parameters as well which enables an easy transition between the three models.
Acoustic shape optimization for various frequencies in different mediums is performed with PSO and the results are compared with the benchmark solutions from the literature for different dimensional problems proving the efficiency of the proposed approach with the following remarks:
- In 2D problems, two BEM methods are used: the conventional isogeometric boundary element method (IGABEM) and the eXtended IGABEM (XIBEM) enriched with the partition-of-unity expansion using a set of plane waves, where the results are generally in good agreement with the linterature with some computation advantage to XIBEM which allows coarser meshes.
-In 3D axi-symmetric problems, the three-dimensional problem is simplified in BEM from a surface integral to a combination of two 1D integrals. The first is the line integral similar to a two-dimensional BEM problem. The second integral is performed over the angle of revolution. The discretization is applied only to the former integration. This leads to significant computational savings and, consequently, better treatment for higher frequencies over the full three-dimensional models.
- In fully 3D problems, a detailed comparison between two BEM methods: the conventional boundary integral equation (CBIE) and Burton-Miller (BM) is provided including the computational cost. The proposed models are enhanced with a modified collocation scheme with offsets to Greville abscissae to avoid placing collocation points at the corners. Placing collocation points on smooth surface enables accurate evaluation of normals for BM formulation in addition to straightforward prediction of jump-terms and avoids singularities in $\mathcal{O} (1/r)$ integrals eliminating the need for polar integration. Furthermore, no additional special treatment is required for the hyper-singular integral while collocating on highly distorted elements, such as those containing sphere poles. The obtained results indicate that, CBIE with PSO is a feasible alternative (except for a small number of fictitious frequencies) which is easier to implement. Furthermore, BM presents an outstanding treatment of the complicated geometry of mufflers with internal extended inlet/outlet tube as an interior 3D Helmholtz acoustic problem instead of using mixed or dual BEM.
2016,2
The gradual digitization in the architecture, engineering, and construction industry over the past fifty years led to an extremely heterogeneous software environment, which today is embodied by the multitude of different digital tools and proprietary data formats used by the many specialists contributing to the design process in a construction project. Though these projects become increasingly complex, the demands on financial efficiency and the completion within a tight schedule grow at the same time. The digital collaboration of project partners has been identified as one key issue in successfully dealing with these challenges. Yet currently, the numerous software applications and their respective individual views on the design process severely impede that collaboration.
An approach to establish a unified basis for the digital collaboration, regardless of the existing software heterogeneity, is a comprehensive digital building model contributed to by all projects partners. This type of data management known as building information modeling (BIM) has many benefits, yet its adoption is associated with many difficulties and thus, proceeds only slowly. One aspect in the field of conflicting requirements on such a digital model is the cooperation of architects and structural engineers. Traditionally, these two disciplines use different abstractions of reality for their models that in consequence lead to incompatible digital representations thereof.
The onset of isogeometric analysis (IGA) promised to ease the discrepancy in design and analysis model representations. Yet, that initial focus quickly shifted towards using these methods as a more powerful basis for numerical simulations. Furthermore, the isogeometric representation alone is not capable of solving the model abstraction problem. It is thus the intention of this work to contribute to an improved digital collaboration of architects and engineers by exploring an integrated analysis approach on the basis of an unified digital model and solid geometry expressed by splines. In the course of this work, an analysis framework is developed that utilizes such models to automatically conduct numerical simulations commonly required in construction projects. In essence, this allows to retrieve structural analysis results from BIM models in a fast and simple manner, thereby facilitating rapid design iterations and profound design feedback.
The BIM implementation Industry Foundation Classes (IFC) is reviewed with regard to its capabilities of representing the unified model. The current IFC schema strongly supports the use of redundant model data, a major pitfall in digital collaboration. Additionally, it does not allow to describe the geometry by volumetric splines. As the pursued approach builds upon a unique model for both, architectural and structural design, and furthermore requires solid geometry, necessary schema modifications are suggested.
Structural entities are modeled by volumetric NURBS patches, each of which constitutes an individual subdomain that, with regard to the analysis, is incompatible with the remaining full model. The resulting consequences for numerical simulation are elaborated in this work. The individual subdomains have to be weakly coupled, for which the mortar method is used. Different approaches to discretize the interface traction fields are implemented and their respective impact on the analysis results is evaluated. All necessary coupling conditions are automatically derived from the related geometry model.
The weak coupling procedure leads to a linear system of equations in saddle point form, which, owed to the volumetric modeling, is large in size and, the associated coefficient matrix has, due to the use of higher degree basis functions, a high bandwidth. The peculiarities of the system require adapted solution methods that generally cause higher numerical costs than the standard procedures for symmetric, positive-definite systems do. Different methods to solve the specific system are investigated and an efficient parallel algorithm is finally proposed.
When the structural analysis model is derived from the unified model in the BIM data, it does in general initially not meet the requirements on the discretization that are necessary to obtain sufficiently accurate analysis results. The consequently necessary patch refinements must be controlled automatically to allowfor an entirely automatic analysis procedure. For that purpose, an empirical refinement scheme based on the geometrical and possibly mechanical properties of the specific entities is proposed. The level of refinement may be selectively manipulated by the structural engineer in charge. Furthermore, a Zienkiewicz-Zhu type error estimator is adapted for the use with isogeometric analysis results. It is shown that also this estimator can be used to steer an adaptive refinement procedure.
2013,2
Modern digital material approaches for the visualization and simulation of heterogeneous materials allow to investigate the behavior of complex multiphase materials with their physical nonlinear material response at various scales. However, these computational techniques require extensive hardware resources with respect to computing power and main memory to solve numerically large-scale discretized models in 3D. Due to a very high number of degrees of freedom, which may rapidly be increased to the two-digit million range, the limited hardware ressources are to be utilized in a most efficient way to enable an execution of the numerical algorithms in minimal computation time. Hence, in the field of computational mechanics, various methods and algorithms can lead to an optimized runtime behavior of nonlinear simulation models, where several approaches are proposed and investigated in this thesis.
Today, the numerical simulation of damage effects in heterogeneous materials is performed by the adaption of multiscale methods. A consistent modeling in the three-dimensional space with an appropriate discretization resolution on each scale (based on a hierarchical or concurrent multiscale model), however, still contains computational challenges in respect to the convergence behavior, the scale transition or the solver performance of the weak coupled problems. The computational efficiency and the distribution among available hardware resources (often based on a parallel hardware architecture) can significantly be improved. In the past years, high-performance computing (HPC) and graphics processing unit (GPU) based computation techniques were established for the investigationof scientific objectives. Their application results in the modification of existing and the development of new computational methods for the numerical implementation, which enables to take advantage of massively clustered computer hardware resources. In the field of numerical simulation in material science, e.g. within the investigation of damage effects in multiphase composites, the suitability of such models is often restricted by the number of degrees of freedom (d.o.f.s) in the three-dimensional spatial discretization. This proves to be difficult for the type of implementation method used for the nonlinear simulation procedure and, simultaneously has a great influence on memory demand and computational time.
In this thesis, a hybrid discretization technique has been developed for the three-dimensional discretization of a three-phase material, which is respecting the numerical efficiency of nonlinear (damage) simulations of these materials. The increase of the computational efficiency is enabled by the improved scalability of the numerical algorithms. Consequently, substructuring methods for partitioning the hybrid mesh were implemented, tested and adapted to the HPC computing framework using several hundred CPU (central processing units) nodes for building the finite element assembly. A memory-efficient iterative and parallelized equation solver combined with a special preconditioning technique for solving the underlying equation system was modified and adapted to enable combined CPU and GPU based computations.
Hence, it is recommended by the author to apply the substructuring method for hybrid meshes, which respects different material phases and their mechanical behavior and which enables to split the structure in elastic and inelastic parts. However, the consideration of the nonlinear material behavior, specified for the corresponding phase, is limited to the inelastic domains only, and by that causes a decreased computing time for the nonlinear procedure. Due to the high numerical effort for such simulations, an alternative approach for the nonlinear finite element analysis, based on the sequential linear analysis, was implemented in respect to scalable HPC. The incremental-iterative procedure in finite element analysis (FEA) during the nonlinear step was then replaced by a sequence of linear FE analysis when damage in critical regions occured, known in literature as saw-tooth approach. As a result, qualitative (smeared) crack initiation in 3D multiphase specimens has efficiently been simulated.
2019,3
Turbomachinery plays an important role in many cases of energy generation or conversion. Therefore, turbomachinery is a promising approaching point for optimization in order to increase the efficiency of energy use. In recent years, the use of automated optimization strategies in combination with numerical simulation has become increasingly popular in many fields of engineering. The complex interactions between fluid and solid mechanics encountered in turbomachines on the one hand and the high computational expense needed to calculate the performance on the other hand, have, however, prevented a widespread use of these techniques in this field of engineering. The objective of this work was the development of a strategy for efficient metamodel based optimization of centrifugal compressor impellers. In this context, the main focus is the reduction of the required numerical expense. The central idea followed in this research was the incorporation of preliminary information acquired from low-fidelity computation methods and empirical correlations into the sampling process to identify promising regions of the parameter space. This information was then used to concentrate the numerically expensive high-fidelity computations of the fluid dynamic and structure mechanic performance of the impeller in these regions while still maintaining a good coverage of the whole parameter space. The development of the optimization strategy can be divided into three main tasks. Firstly, the available preliminary information had to be researched and rated. This research identified loss models based on one dimensional flow physics and empirical correlations as the best suited method to predict the aerodynamic performance. The loss models were calibrated using available performance data to obtain a high prediction quality. As no sufficiently exact models for the prediction of the mechanical loading of the impellercould be identified, a metamodel based on finite element computations was chosen for this estimation. The second task was the development of a sampling method which concentrates samples in regions of the parameter space where high quality designs are predicted by the preliminary information while maintaining a good overall coverage. As available methods like rejection sampling or Markov-chain Monte-Carlo methods did not meet the requirements in terms of sample distribution and input correlation, a new multi-fidelity sampling method called “Filtered Sampling“has been developed. The last task was the development of an automated computational workflow. This workflow encompasses geometry parametrization, geometry generation, grid generation and computation of the aerodynamic performance and the structure mechanic loading. Special emphasis was put into the development of a geometry parametrization strategy based on fluid mechanic considerations to prevent the generation of physically inexpedient designs. Finally, the optimization strategy, which utilizes the previously developed tools, was successfully employed to carry out three optimization tasks. The efficiency of the method was proven by the first and second testcase where an existing compressor design was optimized by the presented method. The results were comparable to optimizations which did not take preliminary information into account, while the required computational expense cloud be halved. In the third testcase, the method was applied to generate a new impeller design. In contrast to the previous examples, this optimization featuredlargervariationsoftheimpellerdesigns. Therefore, theapplicability of the method to parameter spaces with significantly varying designs could be proven, too.
2019,4
The underlying goal of this work is to reduce the uncertainty related to thermally induced stress prediction. This is accomplished by considering use of non-linear material behavior, notably path dependent thermal hysteresis behavior in the elastic properties.
Primary novel factors of this work center on two aspects.
1. Broad material characterization and mechanistic material understanding, giving insight into why this class of material behaves in characteristic manners.
2. Development and implementation of a thermal hysteresis material model and its use to determine impact on overall macroscopic stress predictions.
Results highlight microcracking evolution and behavior as the dominant mechanism for material property complexity in this class of materials. Additionally, it was found that for the cases studied, thermal hysteresis behavior impacts relevant peak stress predictions of a heavy-duty diesel particulate filter undergoing a drop-to-idle regeneration by less than ~15% for all conditions tested. It is also found that path independent heating curves may be utilized for a linear solution assumption to simplify analysis.
This work brings forth a newly conceived concept of a 3 state, 4 path, thermally induced microcrack evolution process; demonstrates experimental behavior that is consistent with the proposed mechanisms, develops a mathematical framework that describes the process and quantifies the impact in a real world application space.
2013,4
This thesis presents two new methods in finite elements and isogeometric analysis for structural analysis. The first method proposes an alternative alpha finite element method using triangular elements. In this method, the piecewise constant strain field of linear triangular finite element method models is enhanced by additional strain terms with an adjustable parameter a, which results in an effectively softer stiffness formulation compared to a linear triangular element. In order to avoid the transverse shear locking of Reissner-Mindlin plates analysis the alpha finite element method is coupled with a discrete shear gap technique for triangular elements to significantly improve the accuracy of the standard triangular finite elements.
The basic idea behind this element formulation is to approximate displacements and rotations as in the standard finite element method, but to construct the bending, geometrical and shear strains using node-based smoothing domains. Several numerical examples are presented and show that the alpha FEM gives a good agreement compared to several other methods in the literature.
Second method, isogeometric analysis based on rational splines over hierarchical T-meshes (RHT-splines) is proposed. The RHT-splines are a generalization of Non-Uniform Rational B-splines (NURBS) over hierarchical T-meshes, which is a piecewise bicubic polynomial over a hierarchical
T-mesh. The RHT-splines basis functions not only inherit all the properties of NURBS such as non-negativity, local support and partition of unity but also more importantly as the capability of joining geometric objects without gaps, preserving higher order continuity everywhere and allow local refinement and adaptivity. In order to drive the adaptive refinement, an efficient recovery-based error estimator is employed. For this problem an imaginary surface is defined. The imaginary surface is basically constructed by RHT-splines basis functions which is used for approximation and interpolation functions as well as the construction of the recovered stress components. Numerical investigations prove that the proposed method is capable to obtain results with higher accuracy and convergence rate than NURBS results.
2022,12
Analysis of Functionally Graded Porous Materials Using Deep Energy Method and Analytical Solution
(2022)
Porous materials are an emerging branch of engineering materials that are composed of two elements: One element is a solid (matrix), and the other element is either liquid or gas. Pores can be distributed within the solid matrix of porous materials with different shapes and sizes. In addition, porous materials are lightweight, and flexible, and have higher resistance to crack propagation and specific thermal, mechanical, and magnetic properties. These properties are necessary for manufacturing engineering structures such as beams and other engineering structures. These materials are widely used in solid mechanics and are considered a good replacement for classical materials by many researchers recently. Producing lightweight materials has been developed because of the possibility of exploiting the properties of these materials. Various types of porous material are generated naturally or artificially for a specific application such as bones and foams. Like functionally graded materials, pore distribution patterns can be uniform or non-uniform. Biot’s theory is a well-developed theory to study the behavior of poroelastic materials which investigates the interaction between fluid and solid phases of a fluid-saturated porous medium.
Functionally graded porous materials (FGPM) are widely used in modern industries, such as aerospace, automotive, and biomechanics. These advanced materials have some specific properties compared to materials with a classic structure. They are extremely light, while they have specific strength in mechanical and high-temperature environments. FGPMs are characterized by a gradual variation of material parameters over the volume. Although these materials can be made naturally, it is possible to design and manufacture them for a specific application. Therefore, many studies have been done to analyze the mechanical and thermal properties of FGPM structures, especially beams.
Biot was the pioneer in formulating the linear elasticity and thermoelasticity equations of porous material. Since then, Biot's formulation has been developed in continuum mechanics which is named poroelasticity. There are obstacles to analyzing the behavior of these materials accurately like the shape of the pores, the distribution of pores in the material, and the behavior of the fluid (or gas) that saturated pores. Indeed, most of the engineering structures made of FGPM have nonlinear governing equations. Therefore, it is difficult to study engineering structures by solving these complicated equations.
The main purpose of this dissertation is to analyze porous materials in engineering structures. For this purpose, the complex equations of porous materials have been simplified and applied to engineering problems so that the effect of all parameters of porous materials on the behavior of engineering structure has been investigated.
The effect of important parameters of porous materials on beam behavior including pores compressibility, porosity distribution, thermal expansion of fluid within pores, the interaction of stresses between pores and material matrix due to temperature increase, effects of pore size, material thickness, and saturated pores with fluid and unsaturated conditions are investigated.
Two methods, the deep energy method, and the exact solution have been used to reduce the problem hypotheses, increase accuracy, increase processing speed, and apply these in engineering structures. In both methods, they are analyzed nonlinear and complex equations of porous materials.
To increase the accuracy of analysis and study of the effect of shear forces, Timoshenko and Reddy's beam theories have been used. Also, neural networks such as residual and fully connected networks are designed to have high accuracy and less processing time than other computational methods.
2014,3
Structural vibration control of high-speed railway bridges using tuned mass dampers, semi-active tuned mass dampers, fluid viscous dampers and magnetorheological dampers to reduce resonant structural vibrations is studied. In this work, the addressed main issues include modeling of the dynamic interaction of the structures, optimization of the parameters of the dampers and comparison of their efficiency.
A new approach to optimize multiple tuned mass damper systems on an uncertain model is proposed based on the H-infinity optimization criteria and the DK iteration procedure with norm-bounded uncertainties in frequency domain. The parameters of tuned mass dampers are optimized directly and simultaneously on different modes contributing significantly to the multi-resonant peaks to explore the different possible combinations of parameters. The effectiveness of the present method is also evaluated through comparison with a previous method.
In the case of semi-active tuned mass dampers, an optimization algorithm is derived to control the magnetorheological damper in these semi-active damping systems. The use of the proposed algorithm can generate various combinations of control gains and state variables. This can lead to the improvement of the ability of MR dampers to track the desired control forces. An uncertain model to reduce detuning effects is also considered in this work.
Next, for fluid viscous dampers, in order to tune the optimal parameters of fluid viscous dampers to the vicinity of the exact values, analytical formulae which can include structural damping are developed based on the perturbation method. The proposed formulae can also be considered as an improvement of the previous analytical formulae, especially for bridge beams with large structural damping.
Finally, a new combination of magnetorheological dampers and a double-beam system to improve the performance of the primary structure vibration is proposed. An algorithm to control magnetorheological dampers in this system is developed by using standard linear matrix inequality techniques. Weight functions as a loop shaping procedure are also introduced in the feedback controllers to improve the tracking ability of magnetorheological damping forces. To this end, the effectiveness of magnetorheological dampers controlled by the proposed scheme, along with the effects of the uncertain and time-delay parameters on the models, are evaluated through numerical simulations.
Additionally, a comparison of the dampers based on their performance is also considered in this work.
2022,4
The Finite Element Method (FEM) is widely used in engineering for solving Partial Differential Equations (PDEs) over complex geometries. To this end, it is required to provide the FEM software with a geometric model that is typically constructed in a Computer-Aided Design (CAD) software. However, FEM and CAD use different approaches for the mathematical description of the geometry. Thus, it is required to generate a mesh, which is suitable for FEM, based on the CAD model. Nonetheless, this procedure is not a trivial task and it can be time consuming. This issue becomes more significant for solving shape and topology optimization problems, which consist in evolving the geometry iteratively. Therefore, the computational cost associated to the mesh generation process is increased exponentially for this type of applications.
The main goal of this work is to investigate the integration of CAD and CAE in shape and topology optimization. To this end, numerical tools that close the gap between design and analysis are presented. The specific objectives of this work are listed below:
• Automatize the sensitivity analysis in an isogeometric framework for applications in shape optimization. Applications for linear elasticity are considered.
• A methodology is developed for providing a direct link between the CAD model and the analysis mesh. In consequence, the sensitivity analysis can be performed in terms of the design variables located in the design model.
• The last objective is to develop an isogeometric method for shape and topological optimization. This method should take advantage of using Non-Uniform Rational B-Splines (NURBS) with higher continuity as basis functions.
Isogeometric Analysis (IGA) is a framework designed to integrate the design and analysis in engineering problems. The fundamental idea of IGA is to use the same basis functions for modeling the geometry, usually NURBS, for the approximation of the solution fields. The advantage of integrating design and analysis is two-fold. First, the analysis stage is more accurate since the system of PDEs is not solved using an approximated geometry, but the exact CAD model. Moreover, providing a direct link between the design and analysis discretizations makes possible the implementation of efficient sensitivity analysis methods. Second, the computational time is significantly reduced because the mesh generation process can be avoided.
Sensitivity analysis is essential for solving optimization problems when gradient-based optimization algorithms are employed. Automatic differentiation can compute exact gradients, automatically by tracking the algebraic operations performed on the design variables. For the automation of the sensitivity analysis, an isogeometric framework is used. Here, the analysis mesh is obtained after carrying out successive refinements, while retaining the coarse geometry for the domain design. An automatic differentiation (AD) toolbox is used to perform the sensitivity analysis. The AD toolbox takes the code for computing the objective and constraint functions as input. Then, using a source code transformation approach, it outputs a code for computing the objective and constraint functions, and their sensitivities as well. The sensitivities obtained from the sensitivity propagation method are compared with analytical sensitivities, which are computed using a full isogeometric approach.
The computational efficiency of AD is comparable to that of analytical sensitivities. However, the memory requirements are larger for AD. Therefore, AD is preferable if the memory requirements are satisfied. Automatic sensitivity analysis demonstrates its practicality since it simplifies the work of engineers and designers.
Complex geometries with sharp edges and/or holes cannot easily be described with NURBS. One solution is the use of unstructured meshes. Simplex-elements (triangles and tetrahedra for two and three dimensions respectively) are particularly useful since they can automatically parameterize a wide variety of domains. In this regard, unstructured Bézier elements, commonly used in CAD, can be employed for the exact modelling of CAD boundary representations. In two dimensions, the domain enclosed by NURBS curves is parameterized with Bézier triangles. To describe exactly the boundary of a two-dimensional CAD model, the continuity of a NURBS boundary representation is reduced to C^0. Then, the control points are used to generate a triangulation such that the boundary of the domain is identical to the initial CAD boundary representation. Thus, a direct link between the design and analysis discretizations is provided and the sensitivities can be propagated to the design domain.
In three dimensions, the initial CAD boundary representation is given as a collection of NURBS surfaces that enclose a volume. Using a mesh generator (Gmsh), a tetrahedral mesh is obtained. The original surface is reconstructed by modifying the location of the control points of the tetrahedral mesh using Bézier tetrahedral elements and a point inversion algorithm. This method offers the possibility of computing the sensitivity analysis using the analysis mesh. Then, the sensitivities can be propagated into the design discretization. To reuse the mesh originally generated, a moving Bézier tetrahedral mesh approach was implemented.
A gradient-based optimization algorithm is employed together with a sensitivity propagation procedure for the shape optimization cases. The proposed shape optimization approaches are used to solve some standard benchmark problems in structural mechanics. The results obtained show that the proposed approach can compute accurate gradients and evolve the geometry towards optimal solutions. In three dimensions, the moving mesh approach results in faster convergence in terms of computational time and avoids remeshing at each optimization step.
For considering topological changes in a CAD-based framework, an isogeometric phase-field based shape and topology optimization is developed. In this case, the diffuse interface of a phase-field variable over a design domain implicitly describes the boundaries of the geometry. The design variables are the local values of the phase-field variable. The descent direction to minimize the objective function is found by using the sensitivities of the objective function with respect to the design variables. The evolution of the phase-field is determined by solving the time dependent Allen-Cahn equation.
Especially for topology optimization problems that require C^1 continuity, such as for flexoelectric structures, the isogeometric phase field method is of great advantage. NURBS can achieve the desired continuity more efficiently than the traditional employed functions. The robustness of the method is demonstrated when applied to different geometries, boundary conditions, and material configurations. The applications illustrate that compared to piezoelectricity, the electrical performance of flexoelectric microbeams is larger under bending. In contrast, the electrical power for a structure under compression becomes larger with piezoelectricity.