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Topology optimization in OpenMDAO

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Abstract

Recently, topology optimization has drawn interest from both industry and academia as the ideal design method for additive manufacturing. Topology optimization, however, has a high entry barrier as it requires substantial expertise and development effort. The typical numerical methods for topology optimization are tightly coupled with the corresponding computational mechanics method such as a finite element method and the algorithms are intrusive, requiring an extensive understanding. This paper presents a modular paradigm for topology optimization using OpenMDAO, an open-source computational framework for multidisciplinary design optimization. This provides more accessible topology optimization algorithms that can be non-intrusively modified and easily understood, making them suitable as educational and research tools. This also opens up further opportunities to explore topology optimization for multidisciplinary design problems. Two widely used topology optimization methods—the density-based and level-set methods—are formulated in this modular paradigm. It is demonstrated that the modular paradigm enhances the flexibility of the architecture, which is essential for extensibility.

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Funding

This paper received support from the NASA Transformational Tools and Technologies Project, contract number NNX15AU22A.

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Authors and Affiliations

  1. University of California San Diego, San Diego, CA, USA

    Hayoung Chung, John T. Hwang & H. Alicia Kim

  2. NASA Glenn Research Center, Cleveland, OH, USA

    Justin S. Gray

  3. Cardiff University, Cardiff, Wales

    H. Alicia Kim

Authors
  1. Hayoung Chung
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  2. John T. Hwang
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  3. Justin S. Gray
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  4. H. Alicia Kim
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Corresponding author

Correspondence to Hayoung Chung.

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Replication of results

To comply with Replication of Results and help the reader in using the present work in research and education, the codes of the present work is open to public e codes (https://github.com/chungh6y/openmdao_TopOpt). Instruction for installation and running the program can be found therein.

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The authors declare that they have no conflict of interest.

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Responsible Editor: Jose Herskovits

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Appendix

Appendix

1.1 A.1 Numerical libraries

In this work, the existing finite element and level-set libraries written by C++ are interfaced to OpenMDAO by Cython (Behnel et al. 2008) for an efficient computation. OpenLSTO (open-sourced level-set topology optimization) is a recently published level-set topology optimization suite and in active development by researchers at the University of Cardiff and the University of California San Diego, which is freely available (https://github.com/M2DOLab/OpenLSTO-lite). The program is designed to provide users with easy access to the level-set method. To make a compilation process straightforward and demonstrate its usage within OpenMDAO, we provide the aforementioned OpenMDAO classes and OpenLSTO as a single suite (https://github.com/chungh6y/openmdao_TopOpt/tree/master/OpenLSTO-master). Detailed installation process is provided therein. After compilation, the user may find two libraries (the finite element and the level-set) as a shared library, which can be imported as the Python object. The concise examples of the given libraries along with the comments are found below.

1.1.1 A.1.1 Finite element method

As specified in the Fig. 6, dimensions (lx, ly), number of elements (nelx, nely), and force conditions are specified. Detailed descriptions of functions that specify boundary conditions can also be found in the GitHub repository. To be used within OpenMDAO, the calculation routines for the stiffness matrices and their derivatives give out the entities of sparse matrix, which is built by SciPy.sparse library.


from pyBind import py_FEA ᅟ # Meshing ============================= fem_solver = py_FEA(lx = 160, ly = 80, nelx=80, nely=40, element_order=2) [node, elem elem_dof]= fem_solver.get_mesh() ᅟ # Material ============================= fem_solver.set_material(E=1., nu=.3) ᅟ # Boundary condition ================== coord = np.array([0,0]) tol = np.array([1e-3,1e10]) fem_solver.set_boundary(coord=coord,tol=tol) BCid = fem_solver.get_boundary() ᅟ # External force as specified at Fig. 2 coord = np.array([length_x,length_y/2]) tol = np.array([1,1]) GF_ = fem_solver.set_force(coord = coord, tol = tol, direction = 1, f = -1.0) ᅟ # Stiffness matrix given as the entities of sparse matrix (CSC) nELEM = elem.shape[0] (rows, cols, vals)=fem_solver.compute_K_SIMP (np.ones(nELEM))

1.2 A.1.2 Level-set method

In the present section, a brief usage of the level-set library is illustrated. The method is classified into three categories: (1) geometric property extraction (2) boundary sensitivity evaluation, (3) and the update of ϕ after suboptimization. After the signed distance function is set (set_levelset), based on the current domain, Ersatz model area fraction as well as the geometric properties are evaluated using the discretise function. The boundary sensitivities bptSens are then obtained from the sensitivity module, where the compliance sensitivities are computed at the boundary points by various subroutines including least square method. Note that these values are the ingredients of suboptimization, wherein advection velocities at the boundary points (Bpt_Vel, timestep) are computed. The level-set function is then updated and reinitialized accordingly.


from py_lsmBind import py_LSM from from pyBind import py_Sensitivity ᅟ # LSM initialization (with initial holes) ================ lsm_solver = py_LSM(nelx = nelx, nely = nely, moveLimit = movelimit) hole = np.array([[16, 14, 5], ... , [144, 66, 5]]) lsm_solver.add_holes(locx = list(hole[:,0]), locy = list(hole[:,1]), radius = list (hole[:,2])) lsm_solver.set_levelset() ᅟ # Discretization ========================================= (bpts_xy, areafraction, seglength) = lsm_solver.discretise() (lb2,ub2) = lsm_solver.get_Lambda_Limits() ᅟ ... FEA that gives out structural response u ... ᅟ # Sensitivity module ==================================== pySens = py_Sensitivity(fem_solver, u) bptSens = pySens.compute_boundary_sens(bpts_xy) ᅟ ... suboptimization through the OpenMDAO ... ᅟ # Advection of the sign distance function ================ lsm_solver.advect(Bpt_Vel, timestep) lsm_solver.reinitialise()

1.3 A.1.3 Configuration of the connectivity

For an illustration of the connectivities established between variables of different Components, we present here the code snippet of the Group object of SIMP topology optimization method.

As the Group component initializes before components are called, a number of shared member variables in relation to the finite element meshes (e.g., nNODE, nELEM) as well as optimization parameters (e.g., penalization order p) are defined using metadata.

The numerous Component objects shown in Fig. 4 are added via add_subsystem, and their variables are connected by connect functions. Note that connections are made at the Group level; therefore, the connections can be modified easilywithout intrusive re-programming of the Components.


from openmdao.api import Group, IndepVarComp from pyBind import py_FEA ᅟ # SIMP Group initialization ============================================ class SimpGroup(Group): ᅟ def initialize(self): # Prescribing shared member variables within group self.metadata. declare('fem_solver', type_=py_FEA, required=True)self.metadata.declare('force', type_=np.ndarray, required=True) self.metadata.declare('num_elem_x', type_=int, required=True) self.metadata. declare('num_elem_y', type_=int, required= True) self.metadata.declare('penal', type_= (int, float), required=True) self.metadata. declare('volume_fraction', type_=(int,float), required=True) ᅟ def setup(self): # Where the Connectivity between components are set up fem_solver = self.metadata['fem_solver'] force = self. metadata['force'] num_elem_x = self.metadata ['num_elem_x'] num_elem_y = self.metadata['num_elem_y'] p = self.metadata['penal'] volume_fraction = self.metadata['volume_fraction'] (nodes, elem, elem_dof) = fem_solver.get_mesh() ᅟ (length_x, length_y) = (np.max(nodes[:, 0], 0), np.max(nodes[:, 1], 0)) (num_nodes_x, num_nodes_y) = (num_elem_x + 1, num_elem_y + 1) ᅟ nNODE = num_nodes_x * num_nodes_y nELEM = (num_nodes_x - 1) * (num_nodes_y - 1) nDOF = nNODE * 2 ᅟ # Input component ============================== # Independent variables are defined as outputs of IndepVarComp component comp = IndepVarComp() comp.add_output('force', val=force) comp.add_output('dvs', val=0.5, shape=nELEM) comp.add_design_var('dvs', lower=0.01, upper =1.0) ## Design variables self.add_subsystem ('cInputs', comp) ᅟ self.connect('cInputs.dvs', 'cFiltering.dvs') self.connect('cInputs.rhs', 'cState.rhs') self.connect('cInputs.rhs', 'cObjective.forces') ᅟ # density filter =============================== comp = DensityFilterComp(length_x=length_x, length_y=length_y, num_nodes_x=num_nodes_x, num_nodes_y=num_nodes_y, num_dvs=nELEM, radius=length_x / (float(num_nodes_x) - 1) * 2) self.add_subsystem('cFiltering', comp) self.connect('cFiltering.dvs_bar', 'cPenalty .x') self.connect('cFiltering.dvs_bar', 'cConstraint.x') ᅟ # penalization ================================= comp = PenalizationComp(num=nELEM, p=p) self.add_subsystem('cPenalty', comp) self.connect('cPenalty.y', 'cState.multipliers') ᅟ # states ======================================= comp = StatesComp(fem_solver=fem_solver, num_nodes_x=num_nodes_x, num_nodes_y= num_nodes_y, isSIMP = True) self.add_subsystem('cState', comp) self.connect('cState.states', 'cObjective.disp') ᅟ # compliance =================================== comp = ComplianceComp(num_nodes_x=num_nodes_x, num_nodes_y=num_nodes_y) comp.add_objective ('compliance') ## Objective self.add_subsystem('cObjective', comp) ᅟ # weight ======================================= comp = WeightComp(num=nELEM) comp.add_constraint ('weight', upper=volume_fraction) ## Constraint self.add_subsystem('cConstraint', comp)

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Chung, H., Hwang, J.T., Gray, J.S. et al. Topology optimization in OpenMDAO. Struct Multidisc Optim 59, 1385–1400 (2019). https://doi.org/10.1007/s00158-019-02209-7

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