Numerical solution of solidification and elastodynamics problems using dynamic substructuring based on adaptive error estimation

The classical finite element method (FEM) is one of the most powerful tools for solution of engineering problems. Main idea of FEM is to use polynomials as approximation functions and due to this fact that solutions obtained by FEM have smooth characteristic. When we consider the dynamic problems that contain moving discontinuities such as solidification problems and elastodynamics problems, very fine meshes need to be used to obtain acceptable results because of the smooth characteristic of FEM. But, using very fine mesh in computational domain increases CPU times and solution costs. This is the main disadvantage of classical FEM. In this thesis, a new technique called as dynamic substructuring based on adaptive error estimation is developed and applied to solidification problems and elastodynamics problems to overcome solution cost disadvantage of FEM. When the solidification problems are examined two points need to be cared to be able to obtain accurate results. One of them is capturing the correct location of solidification front and other point is capturing the zones affected by solidification front such as mushy zone. When the previously developed methods to overcome solution cost disadvantage of FEM are examined it is observed that moving mesh methods, generalized FEM (GFEM), eXtended FEM (XFEM) and meshless methods focus on aligning the nodes to the discontinuity interface. Thus, very accurate solutions around the discontinuity interface can be obtained with these methods. However, all of these methods ignore mushy zone during the solution of solidification problems. On the other hand, error estimator based adaptive remeshing methods are also used for solution of solidification problems. Unlike other methods this method can also capture both solidification front and the mushy zone but disadvantage of it is high CPU times because the whole domain is remeshed with refined elements in these methods. When the size of model geometry increases, the solution time also increases drastically. Another disadvantage of this method is mesh compatibility. In this method during remeshing, old mesh data needs to be mapped onto new mesh data and special algorithms are required to obtain accurate solutions. If the local remeshing is applied to the domain, it is hard to connect the new generated refined elements to the coarse mesh of domain. This phenomenon does not occur at dynamic substructuring based on adaptive error estimation approach because the substructure is created explicitly and independent from the original global model. Thus, substructure domain elements can be refined freely. When the elastodynamics problems are examined three points need to be cared to be able to obtain accurate results. One of them is capturing correct location of wave front, other point is capturing wave length region and last point is reducing dissipation and dispersion errors. While the dissipation errors are caused by large time steps, dispersion errors are caused by large element sizes. Most straightforwardly, a fine mesh with a very small time step size can be used to get accurate results of elastodynamics problems. But this condition causes CPU times and solution costs to increase drastically. There are some methods to get optimal accuracy while decreasing solution times in literature. GFEM / XFEM / enriched FEM and meshless methods are very powerful methods and they are used for obtaining accurate results at discontinuity interface. These methods adds imaginary or real nodes to discontinuity interface but they ignore elements located in wave length region. So, dispersion errors are ignored in wave length region. Spectral FEM and wavelet FEM methods use special approximation functions to obtain accurate results in computational domain however use of these functions is computationally expensive. Space-time FEM method uses finite element discretizations in space and time simultaneously with basis functions which are continuous in space and discontinuous at the discrete time levels. Good convergence and stability properties near sharp gradients are obtained in solution. But, CPU time and solution costs are high due to complex formulation and large sizes of linear systems. This condition is a disadvantage for solution of problems which have large domains. Error estimator based adaptive remeshing methods and adaptive scaled boundary FEM methods are also used for solution of elastodynamics problems. When these methods are applied to elastodynamics problems they have the same advantages and disadvantages with solidification problems told in previous paragraph. Adaptive mesh superposition FEM method creates independent refined subdomains and refines the mesh locally by superposition without changing global matrices. This method enables to use fine mesh around captured locations and to get accurate results around these locations. These properties of adaptive mesh superposition finite element method seems very similar with the dynamic substructuring approach based on adaptive error estimation which is proposed in this thesis. However there are certain differences between two methods. While the adaptive mesh superposition method obtains the solutions by superposing local refined mesh results with global coarse mesh results, dynamic substructuring approach based on adaptive error estimation method obtains the local refined mesh results independent from global coarse mesh results and augments obtained local fine mesh results to global coarse mesh results. When the local mesh node and global mesh node coincides, adaptive mesh superposition FEM supposes local displacement values as zero to ensure compatibility of the total displacement solution and uniqueness of the displacement fields on global mesh and local mesh. So, errors on nodal solutions of coincident nodes are not improved with adaptive mesh superposition FEM. Dynamic substructuring approach based on adaptive error estimation method uses nodes of elements which are neighbor to elements that have larger error levels than permissible error threshold as boundary condition. So, nodal solutions of all error elements are improved. So, more accurate results can be obtained with proposed dynamic substructuring approach based on adaptive error estimation method. Adaptive Mesh superposition FEM uses local coordinates to create coupling matrix and it uses inverse mapping and newton-Raphson iterations to transfer from global coordinates to local coordinates. These extra calculations causes solution times to increase. Dynamic substructuring approach based on adaptive error estimation method augments the local solution to global solution. So, inverse mapping and Newton-Raphson iterations are not used. So, proposed method in this thesis works faster than adaptive mesh superposition FEM. Motivated by the drawbacks of above methods, a new technique using FEM is proposed called dynamic substructuring based on adaptive error estimation. In this method, the original FEM mesh is not changed during the solution steps and accurate solutions can be obtained using very coarse meshes. The computational domain is meshed with a coarse mesh and elements having errors above a threshold value which includes solidification front, wave front, mushy zone and wave length zone are captured with an error energy norm estimator at each analysis step. Then, a substructure independent from the global model is generated by refining the elements having errors above the error threshold; thus, remeshing of the entire original mesh at each analysis step is avoided that helps reduce the CPU time considerably. Substructure boundary conditions are obtained from the nodes of global model which are the neighbours to the elements used for the substructure. Following, the original global equations are augmented by the substructure equations and then the coupled equations of original and substructure equations are solved simultaneously. The mesh refinement is adaptive based on the adaptive error estimator calculations and the refined regions are removed in the substructure if their error level reduces to below the threshold value during analysis steps.. In solidification chapter of this thesis, developed method is applied to solidification of lead alloy. Firstly, benchmark studies are performed to examine the reliability of developed Matlab code by applying it a simple transient heat transfer problem and obtained results are compared with results of same problem modelled with commercial Abaqus software. After getting accurate results with benchmark study, developed dynamic substructuring method code is applied to solidification problem. Superconvergent patch recovery technique is used to obtain the heat fluxes to evaluate the error energy norm of elements at each analysis step. Solidification front, mushy zone and elements having errors above a threshold value are captured with the error estimator. Then, elements having errors above the threshold value are refined by creating a substructure which is independent from the original global mesh. Equations of the global coarse mesh are augmented with the equations of the substructure and solutions are obtained. Same problem is also solved with fine mesh with classical FEM. Results and solution times of classical FEM with fine mesh and dynamic substructuring method with coarse mesh are compared with each other. It is seen that employing the equations of the original coarse mesh help reduce the computational cost. Numerical solutions are presented and it is shown that the proposed approach has advantages over the alternative methods and, by the virtue of the adaptive error estimation algorithm, significantly decreases the CPU time of numerical solutions while it increases the accuracy of solutions and locates precisely the solidification front and mushy zone. In elastodynamics chapter of this thesis, developed method is applied to an elastodynamics problem. Firstly, benchmark studies are performed to examine the reliability of developed Matlab code by applying it a simple problem and obtained results are compared with results of same problem modelled with commercial Abaqus software. After getting accurate results with benchmark study, developed dynamic substructuring method based on error estimation code applied to elastodynamics problem. Superconvergent patch recovery technique is used to obtain the stresses to evaluate the error energy norm of elements at each analysis step. Stress wave front, stress wave length zone and elements having errors above a threshold value are captured with the error estimator. Then, elements having errors above the threshold value are refined by creating a substructure which is independent from the original global mesh. Equations of the global coarse mesh are augmented with the equations of the substructure and solutions are obtained. Same problem is also solved with fine mesh with classical FEM. Results and solution times of classical FEM with fine mesh and dynamic substructuring method with coarse mesh are compared with each other. It is seen that employing the equations of the original coarse mesh help reduce the computational cost. Numerical solutions are presented and it is shown that the proposed approach has advantages over the alternative methods and, by the virtue of the adaptive error estimation algorithm, significantly decreases the CPU time of numerical solutions while it increases the accuracy of solutions and locates precisely the stress wave front and stress wave length zone. Under the light of numerical results, it is observed that dynamic substructuring based on adaptive error estimation technique is more advantageous when compared with the other methods in literature whose main drawbacks are listed below. There is no clear CPU time comparisions of alternative methods in literature. But, when the alternative methods are considered, most of the methods require that node numbers and element numbers of the global domain are to be increased to improve the accuracy and this condition causes the global matrices to be updated at each analysis step. When the computation domain gets larger, solution time will increase drastically. Advantage of dynamic substructuring based on adaptive error estimation method arises at this point. The original global model is not affected when dynamic substructuring based on adaptive error estimation approach is used. The substructure which is a smaller portion of the global model is prepared at each step. Thus, CPU times for substructure matrix preparation will be less time consuming when compared with alternative methods. Moreover, the initial global equations can be used throughout the solution procedure if associated FEM matrices are unchanged that further reduces the CPU time of the proposed approach. Other methods which do not require remeshing use higher order elements and complex solution methodologies. Complexity of solution methodologies and use of higher order elements causes solution times to increase. Dynamic substructuring based on adaptive error estimation method uses classical finite element formulations which are simple and easy to imlement. So, computational costs of these elements are lower than alternative methods in literature.