Chandrashekhar Jog - Profile on Academia.edu (original) (raw)
Papers by Chandrashekhar Jog
Fluids, 2022
This work develops a new monolithic finite-element-based strategy for magnetohydrodynamics (MHD) ... more This work develops a new monolithic finite-element-based strategy for magnetohydrodynamics (MHD) involving a compressible fluid based on a continuous velocity–pressure formulation. The entire formulation is within a nodal finite element framework, and is directly in terms of physical variables. The exact linearization of the variational formulation ensures a quadratic rate of convergence in the vicinity of the solution. Both steady-state and transient formulations are presented for two- and three-dimensional flows. Several benchmark problems are presented, and comparisons are carried out against analytical solutions, experimental data, or against other numerical schemes for MHD. We show a good coarse-mesh accuracy and robustness of the proposed strategy, even at high Hartmann numbers.
Topology Design of Three-Dimensional Structures Using Hybrid Finite Elements
Solid Mechanics and Its Applications
ABSTRACT Conventional three-dimensional isoparametric elements are susceptible to problems of loc... more ABSTRACT Conventional three-dimensional isoparametric elements are susceptible to problems of locking when used to model plate/shell geometries or when the meshes are distorted etc. Hybrid elements that are based on a two-field variational formulation are immune to most of these problems, and hence can be used to efficiently model both “chunky” three-dimensional and plate/shell type structures. Thus, only one type of element can be used to model “all” types of structures, and also allows us to use a standard dual algorithm for carrying out the topology optimization of the structure. We also address the issue of manufacturability of the designs.
Mixed finite elements for electromagnetic analysis
Computers & Mathematics with Applications, 2014
ABSTRACT The occurrence of spurious solutions is a well-known limitation of the standard nodal fi... more ABSTRACT The occurrence of spurious solutions is a well-known limitation of the standard nodal finite element method when applied to electromagnetic problems. The two commonly used remedies that are used to address this problem are (i) The addition of a penalty term with the penalty factor based on the local dielectric constant, and which reduces to a Helmholtz form on homogeneous domains (regularized formulation); (ii) A formulation based on a vector and a scalar potential. Both these strategies have some shortcomings. The penalty method does not completely get rid of the spurious modes, and both methods are incapable of predicting singular eigenvalues in non-convex domains. Some non-zero spurious eigenvalues are also predicted by these methods on non-convex domains. In this work, we develop mixed finite element formulations which predict the eigenfrequencies (including their multiplicities) accurately, even for nonconvex domains. The main feature of the proposed mixed finite element formulation is that no ad-hoc terms are added to the formulation as in the penalty formulation, and the improvement is achieved purely by an appropriate choice of finite element spaces for the different variables. We show that the formulation works even for inhomogeneous domains where ‘double noding’ is used to enforce the appropriate continuity requirements at an interface. For two-dimensional problems, the shape of the domain can be arbitrary, while for the three-dimensional ones, with our current formulation, only regular domains (which can be nonconvex) can be modeled. Since eigenfrequencies are modeled accurately, these elements also yield accurate results for driven problems.
A Note on Boundary Conditions
Foundations and Applications of Mechanics
Some Results in n-Dimensional Euclidean Spaces
Foundations and Applications of Mechanics
Oblate Spheroidal Coordinate System
Foundations and Applications of Mechanics
Nonlinear Elasticity
Foundations and Applications of Mechanics
Constitutive Equations
Foundations and Applications of Mechanics
Introduction to Tensors
Foundations and Applications of Mechanics
Structural Optimization, 1996
This paper introduces a method for variabletopology shape optimization of elastic structures call... more This paper introduces a method for variabletopology shape optimization of elastic structures called the perimeter method. An upper-bound constraint on the perimeter of the solid part of the structure ensures a well-posed design problem. The perimeter constraint allows the designer to control the number of holes in the optimal design and to establish their characteristic length scale. Finite element implementations generate practical designs that are convergent with respect to grid refinement. Thus, an arbitrary level of geometric resolution can be achieved, so single-step procedures for topology design and detailed shape design are possible. The perimeter method eliminates the need for relaxation, thereby circumventing many of the complexities and restrictions of other approaches to topology design. and Strang 1986). In particular, we typically can construct a nonconvergent sequence such that the compliance reduces monotonically. Therefore, an optimal design does not exist.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
We consider the synchronous whirl of arbitrary axisymmetric rotors supported on rigid bearings. P... more We consider the synchronous whirl of arbitrary axisymmetric rotors supported on rigid bearings. Prior computational treatments of this problem were based on adding element-level gyroscopic terms to the governing equations. Here, we begin with a direct continuum formulation wherein gyroscopic terms need not be added on separately and explicitly: all gyroscopic effects are captured implicitly within the continuum elastodynamics. We present two new methods for obtaining the whirl speed, where we project the dynamic equilibrium equations of the rotor on to a few of its non-spinning vibration mode shapes. The first modal projection method is direct and more accurate, but requires numerical evaluation of more demanding integrals. The second method is iterative and involves a small approximation, but is simpler. Both the methods are based on one new insight: the gyroscopic terms used in other treatments are essentially the result of a prestress in the rotor caused by the non-zero spin rate...
Symmetry-breaking transitions in equilibrium shapes of coherent precipitates
Journal of the Mechanics and Physics of Solids, 2000
We present a general approach for determining the equilibrium shape of isolated, coherent, misfit... more We present a general approach for determining the equilibrium shape of isolated, coherent, misfitting particles by minimizing the sum of elastic and interfacial energies using a synthesis of finite element and optimization techniques. The generality derives from the fact that there is no ...
The Journal of the Acoustical Society of America, 2005
Journal of Sound and Vibration, 2002
Although a lot of attention in the topology optimization literature has focused on the optimizati... more Although a lot of attention in the topology optimization literature has focused on the optimization of eigenfrequencies in free vibration problems, relatively little work has been done on the optimization of structures subjected to periodic loading. In this paper, we propose two measures, one global and the other local, for the minimization of vibrations of structures subjected to periodic loading. The global measure which we term as the &&dynamic compliance'' reduces the vibrations in an overall sense, and thus has important implications from the viewpoint of reducing the noise radiated from a structure, while the local measure reduces the vibrations at a user-de"ned point. Both measures bring about a reduction in the vibration level by moving the natural frequencies which contribute most signi"cantly to the measures, away from the driving frequencies, although, as expected, in di!erent ways. Quite surprisingly, the structure of the dynamic compliance optimization problem turns out to be very similar to the structure of the static compliance optimization problem. The availability of analytical sensitivities results in an e$cient algorithm for both measures. We show the e!ectiveness of the measures by presenting some numerical examples. 2002 Elsevier Science Ltd. All rights reserved. Using the expression for K given by equation ( ) in conjunction with equation ( ), we get *K * G " * * G (K! ; M)C\(K! ; M)!(K! ; M)C\ *C * G C\(K! ( M) #(K! ; M)C\ * * G (K! ; M)# ( *C * G .
Journal of Elasticity, 2008
The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials.... more The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials. Here we show how the logarithm of an arbitrary tensor can be explicitly evaluated for any underlying space dimension n. We also present a method for the explicit evaluation of the derivatives of the logarithm of a tensor.
Shortcomings of discontinuous-pressure finite element methods on a class of transient problems
International Journal for Numerical Methods in Fluids, 2009
ABSTRACT Past studies that have compared LBB stable discontinuous- and continuous-pressure finite... more ABSTRACT Past studies that have compared LBB stable discontinuous- and continuous-pressure finite element formulations on a variety of problems have concluded that both methods yield solutions of comparable accuracy, and that the choice of interpolation is dictated by which of the two is more efficient. In this work, we show that using discontinuous-pressure interpolations can yield inaccurate solutions at large times on a class of transient problems, while the continuous-pressure formulation yields solutions that are in good agreement with the analytical solution. Copyright © 2009 John Wiley & Sons, Ltd.
A finite element method for compressible viscous fluid flows
International Journal for Numerical Methods in Fluids, 2010
ABSTRACT This work presents a mixed three-dimensional finite element formulation for analyzing co... more ABSTRACT This work presents a mixed three-dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady-state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering, 1994
Significant performance improvements can be obtained if the topology of an elastic structure is a... more Significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems. We study the optimal shape design of a two-dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material. The macroscopic version of this problem is not well-posed if no restrictions are placed on the structure topology; relaxation of the optimization problem via quasiconvexification or homogenization methods is required. The effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material. A combined analyticalcomputational approach is proposed to solve the relaxed optimization problem. Both stress and displacement analysis methods are presented. Since rank-2 layered composites are known to achieve optimal energy bounds, we restrict the design space to this class of microstructures whose effective properties can easily be determined in explicit form. We develop a series of reduced problems by sequentially interchanging extremization operators and analytically optimizing the microstructural design fields. This results in optimization problems involving the distribution of an adaptive material that continuously optimizes its microstructure in response to the current state of stress or strain. A further reduced problem, involving only the response field, can be obtained in the stress-based approach, but the requisite interchange of extremization operators is not valid in the case of the displacement-based model. Finite element optimization procedures based on the reduced displacement formulation are developed and numerical solutions are presented. Care must be taken in selecting the discrete function spaces for the design density and displacement response, since the reduced problem is a two-field, mixed variational problem. An improper choice for the solution space leads to instabilities in the optimal design similar to those encountered in mixed formulations of the Stokes problem. * Most authors would not classify these as shape optimization problems, but we mention them here because they do involve variations in the three-dimensional geometry of the structure
Distributed-parameter optimization and topology design for non-linear thermoelasticity
Computer Methods in Applied Mechanics and Engineering, 1996
... in the presence of thermal loading, the stresses are not proportional to the strains, even in... more ... in the presence of thermal loading, the stresses are not proportional to the strains, even in linearized thermoelasticity. ... This result is analogous to that obtained using the small-strain theory, hence, it would be reasonable to expect that the solid-void compliance optimization ...
Higher-order shell elements based on a Cosserat model, and their use in the topology design of structures
Computer Methods in Applied Mechanics and Engineering, 2004
For the purpose of carrying out topology optimization of shell structures, we show that it is adv... more For the purpose of carrying out topology optimization of shell structures, we show that it is advantageous to use shell elements based on a classical shell theory (such as the Cosserat theory) rather than elements based on the degenerated solid approach, since the shell thickness appears explicitly in the formulation, thereby greatly simplifying the sensitivity analysis. One of the well-known shell elements based on the Cosserat shell theory is the four-node element presented by Simo et al. [Comput. Methods Appl. Mech. Engrg. 72 (1989) 267; 73 (1989) 53]. Although one could use this element, the use of lower-order elements often results in instabilities (such as the “checkerboard” instability) in the resulting topologies; this provides the motivation for developing higher-order shell elements based on a classical shell theory. In this work, we present the formulation and implementation details for six-node and seven-node triangular, and nine-node quadrilateral shell elements, which are based on the variational formulation of Simo et al., and show that good accuracy is obtained even with coarse meshes in fairly demanding problems. We also present the topology optimization formulation, and some examples of optimal topologies that are obtained using this formulation.
Fluids, 2022
This work develops a new monolithic finite-element-based strategy for magnetohydrodynamics (MHD) ... more This work develops a new monolithic finite-element-based strategy for magnetohydrodynamics (MHD) involving a compressible fluid based on a continuous velocity–pressure formulation. The entire formulation is within a nodal finite element framework, and is directly in terms of physical variables. The exact linearization of the variational formulation ensures a quadratic rate of convergence in the vicinity of the solution. Both steady-state and transient formulations are presented for two- and three-dimensional flows. Several benchmark problems are presented, and comparisons are carried out against analytical solutions, experimental data, or against other numerical schemes for MHD. We show a good coarse-mesh accuracy and robustness of the proposed strategy, even at high Hartmann numbers.
Topology Design of Three-Dimensional Structures Using Hybrid Finite Elements
Solid Mechanics and Its Applications
ABSTRACT Conventional three-dimensional isoparametric elements are susceptible to problems of loc... more ABSTRACT Conventional three-dimensional isoparametric elements are susceptible to problems of locking when used to model plate/shell geometries or when the meshes are distorted etc. Hybrid elements that are based on a two-field variational formulation are immune to most of these problems, and hence can be used to efficiently model both “chunky” three-dimensional and plate/shell type structures. Thus, only one type of element can be used to model “all” types of structures, and also allows us to use a standard dual algorithm for carrying out the topology optimization of the structure. We also address the issue of manufacturability of the designs.
Mixed finite elements for electromagnetic analysis
Computers & Mathematics with Applications, 2014
ABSTRACT The occurrence of spurious solutions is a well-known limitation of the standard nodal fi... more ABSTRACT The occurrence of spurious solutions is a well-known limitation of the standard nodal finite element method when applied to electromagnetic problems. The two commonly used remedies that are used to address this problem are (i) The addition of a penalty term with the penalty factor based on the local dielectric constant, and which reduces to a Helmholtz form on homogeneous domains (regularized formulation); (ii) A formulation based on a vector and a scalar potential. Both these strategies have some shortcomings. The penalty method does not completely get rid of the spurious modes, and both methods are incapable of predicting singular eigenvalues in non-convex domains. Some non-zero spurious eigenvalues are also predicted by these methods on non-convex domains. In this work, we develop mixed finite element formulations which predict the eigenfrequencies (including their multiplicities) accurately, even for nonconvex domains. The main feature of the proposed mixed finite element formulation is that no ad-hoc terms are added to the formulation as in the penalty formulation, and the improvement is achieved purely by an appropriate choice of finite element spaces for the different variables. We show that the formulation works even for inhomogeneous domains where ‘double noding’ is used to enforce the appropriate continuity requirements at an interface. For two-dimensional problems, the shape of the domain can be arbitrary, while for the three-dimensional ones, with our current formulation, only regular domains (which can be nonconvex) can be modeled. Since eigenfrequencies are modeled accurately, these elements also yield accurate results for driven problems.
A Note on Boundary Conditions
Foundations and Applications of Mechanics
Some Results in n-Dimensional Euclidean Spaces
Foundations and Applications of Mechanics
Oblate Spheroidal Coordinate System
Foundations and Applications of Mechanics
Nonlinear Elasticity
Foundations and Applications of Mechanics
Constitutive Equations
Foundations and Applications of Mechanics
Introduction to Tensors
Foundations and Applications of Mechanics
Structural Optimization, 1996
This paper introduces a method for variabletopology shape optimization of elastic structures call... more This paper introduces a method for variabletopology shape optimization of elastic structures called the perimeter method. An upper-bound constraint on the perimeter of the solid part of the structure ensures a well-posed design problem. The perimeter constraint allows the designer to control the number of holes in the optimal design and to establish their characteristic length scale. Finite element implementations generate practical designs that are convergent with respect to grid refinement. Thus, an arbitrary level of geometric resolution can be achieved, so single-step procedures for topology design and detailed shape design are possible. The perimeter method eliminates the need for relaxation, thereby circumventing many of the complexities and restrictions of other approaches to topology design. and Strang 1986). In particular, we typically can construct a nonconvergent sequence such that the compliance reduces monotonically. Therefore, an optimal design does not exist.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
We consider the synchronous whirl of arbitrary axisymmetric rotors supported on rigid bearings. P... more We consider the synchronous whirl of arbitrary axisymmetric rotors supported on rigid bearings. Prior computational treatments of this problem were based on adding element-level gyroscopic terms to the governing equations. Here, we begin with a direct continuum formulation wherein gyroscopic terms need not be added on separately and explicitly: all gyroscopic effects are captured implicitly within the continuum elastodynamics. We present two new methods for obtaining the whirl speed, where we project the dynamic equilibrium equations of the rotor on to a few of its non-spinning vibration mode shapes. The first modal projection method is direct and more accurate, but requires numerical evaluation of more demanding integrals. The second method is iterative and involves a small approximation, but is simpler. Both the methods are based on one new insight: the gyroscopic terms used in other treatments are essentially the result of a prestress in the rotor caused by the non-zero spin rate...
Symmetry-breaking transitions in equilibrium shapes of coherent precipitates
Journal of the Mechanics and Physics of Solids, 2000
We present a general approach for determining the equilibrium shape of isolated, coherent, misfit... more We present a general approach for determining the equilibrium shape of isolated, coherent, misfitting particles by minimizing the sum of elastic and interfacial energies using a synthesis of finite element and optimization techniques. The generality derives from the fact that there is no ...
The Journal of the Acoustical Society of America, 2005
Journal of Sound and Vibration, 2002
Although a lot of attention in the topology optimization literature has focused on the optimizati... more Although a lot of attention in the topology optimization literature has focused on the optimization of eigenfrequencies in free vibration problems, relatively little work has been done on the optimization of structures subjected to periodic loading. In this paper, we propose two measures, one global and the other local, for the minimization of vibrations of structures subjected to periodic loading. The global measure which we term as the &&dynamic compliance'' reduces the vibrations in an overall sense, and thus has important implications from the viewpoint of reducing the noise radiated from a structure, while the local measure reduces the vibrations at a user-de"ned point. Both measures bring about a reduction in the vibration level by moving the natural frequencies which contribute most signi"cantly to the measures, away from the driving frequencies, although, as expected, in di!erent ways. Quite surprisingly, the structure of the dynamic compliance optimization problem turns out to be very similar to the structure of the static compliance optimization problem. The availability of analytical sensitivities results in an e$cient algorithm for both measures. We show the e!ectiveness of the measures by presenting some numerical examples. 2002 Elsevier Science Ltd. All rights reserved. Using the expression for K given by equation ( ) in conjunction with equation ( ), we get *K * G " * * G (K! ; M)C\(K! ; M)!(K! ; M)C\ *C * G C\(K! ( M) #(K! ; M)C\ * * G (K! ; M)# ( *C * G .
Journal of Elasticity, 2008
The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials.... more The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials. Here we show how the logarithm of an arbitrary tensor can be explicitly evaluated for any underlying space dimension n. We also present a method for the explicit evaluation of the derivatives of the logarithm of a tensor.
Shortcomings of discontinuous-pressure finite element methods on a class of transient problems
International Journal for Numerical Methods in Fluids, 2009
ABSTRACT Past studies that have compared LBB stable discontinuous- and continuous-pressure finite... more ABSTRACT Past studies that have compared LBB stable discontinuous- and continuous-pressure finite element formulations on a variety of problems have concluded that both methods yield solutions of comparable accuracy, and that the choice of interpolation is dictated by which of the two is more efficient. In this work, we show that using discontinuous-pressure interpolations can yield inaccurate solutions at large times on a class of transient problems, while the continuous-pressure formulation yields solutions that are in good agreement with the analytical solution. Copyright © 2009 John Wiley & Sons, Ltd.
A finite element method for compressible viscous fluid flows
International Journal for Numerical Methods in Fluids, 2010
ABSTRACT This work presents a mixed three-dimensional finite element formulation for analyzing co... more ABSTRACT This work presents a mixed three-dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady-state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering, 1994
Significant performance improvements can be obtained if the topology of an elastic structure is a... more Significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems. We study the optimal shape design of a two-dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material. The macroscopic version of this problem is not well-posed if no restrictions are placed on the structure topology; relaxation of the optimization problem via quasiconvexification or homogenization methods is required. The effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material. A combined analyticalcomputational approach is proposed to solve the relaxed optimization problem. Both stress and displacement analysis methods are presented. Since rank-2 layered composites are known to achieve optimal energy bounds, we restrict the design space to this class of microstructures whose effective properties can easily be determined in explicit form. We develop a series of reduced problems by sequentially interchanging extremization operators and analytically optimizing the microstructural design fields. This results in optimization problems involving the distribution of an adaptive material that continuously optimizes its microstructure in response to the current state of stress or strain. A further reduced problem, involving only the response field, can be obtained in the stress-based approach, but the requisite interchange of extremization operators is not valid in the case of the displacement-based model. Finite element optimization procedures based on the reduced displacement formulation are developed and numerical solutions are presented. Care must be taken in selecting the discrete function spaces for the design density and displacement response, since the reduced problem is a two-field, mixed variational problem. An improper choice for the solution space leads to instabilities in the optimal design similar to those encountered in mixed formulations of the Stokes problem. * Most authors would not classify these as shape optimization problems, but we mention them here because they do involve variations in the three-dimensional geometry of the structure
Distributed-parameter optimization and topology design for non-linear thermoelasticity
Computer Methods in Applied Mechanics and Engineering, 1996
... in the presence of thermal loading, the stresses are not proportional to the strains, even in... more ... in the presence of thermal loading, the stresses are not proportional to the strains, even in linearized thermoelasticity. ... This result is analogous to that obtained using the small-strain theory, hence, it would be reasonable to expect that the solid-void compliance optimization ...
Higher-order shell elements based on a Cosserat model, and their use in the topology design of structures
Computer Methods in Applied Mechanics and Engineering, 2004
For the purpose of carrying out topology optimization of shell structures, we show that it is adv... more For the purpose of carrying out topology optimization of shell structures, we show that it is advantageous to use shell elements based on a classical shell theory (such as the Cosserat theory) rather than elements based on the degenerated solid approach, since the shell thickness appears explicitly in the formulation, thereby greatly simplifying the sensitivity analysis. One of the well-known shell elements based on the Cosserat shell theory is the four-node element presented by Simo et al. [Comput. Methods Appl. Mech. Engrg. 72 (1989) 267; 73 (1989) 53]. Although one could use this element, the use of lower-order elements often results in instabilities (such as the “checkerboard” instability) in the resulting topologies; this provides the motivation for developing higher-order shell elements based on a classical shell theory. In this work, we present the formulation and implementation details for six-node and seven-node triangular, and nine-node quadrilateral shell elements, which are based on the variational formulation of Simo et al., and show that good accuracy is obtained even with coarse meshes in fairly demanding problems. We also present the topology optimization formulation, and some examples of optimal topologies that are obtained using this formulation.