1.  Speaker: Nguyen Xuan Hung:

Title: Topology optimization based on polytopal elements for additive manufacturing

Abstract:

Polytopal elements, including polygonal elements for a two-dimensional case and polyhedral elements for a three-dimensional case, are adopted to mesh material domains with high complexity. In our recent work [1], polytopal composite finite elements (PCEs), which were proposed to model the mechanical behavior of compressible and incompressible materials, could pass the patch test and satisfy the inf-sup stability with high accuracy. The PCEs were constructed with a polynomial projection of compatible strain fields through the least-squares approximation. For incompressible materials, volumetric locking arises when the Poison’s ratio close to being 0.5, leading to the over stiffening of incompressible material elements. The volumetric locking phenomenon occurs when a fully integrated element (e.g., Q4 elements with 4 quadrature points or H8 elements with 8 quadrature points) is employed. One well-known solution to the volumetric locking is to couple pressure unknowns together with displacement unknowns. However, this way requires additional unknowns in the system of linear equations, thus require large storage and cost expensively. Fortunately, PCEs, which were purely based on displacement formulation without the additional unknowns, could overcome the over stiffening to accurately model incompressible materials.

The moving morphable bar (MMB) approach [2], which has been proposed for structural optimization recently, has advantages: (1) using a small number of design variables; (2) accurately capturing structural-boundary features; (3) easy control of structural feature sizes; and (4) direct multiscale design of porous structures without material homogenization, connector constraints, and local volume constraints [3]. In this study, we present an efficient computational tool using a combination of PCEs and MMBs to model and design incompressible materials (i.e., rubbers), but it can be extended to other material models. Especially, the presentation emphasizes the direct multiscale design of porous structures (i.e., biology soft tissues) for additive manufacturing techniques. The idea is to project adaptive geometric components (AGCs) onto a fixed mesh of PCEs. The structure is simultaneously optimized at both macro-structural and micro-structural scales by directly optimizing geometry parameters of AGCs. Some numerical examples are investigated to verify the effectiveness of the proposed approach.

References

[1] H. Nguyen-Xuan, K.N. Chau, K.N. Chau, Polytopal composite finite elements, Comput. Methods Appl. Mech. Eng. 355 (2019) 405–437. doi:10.1016/j.cma.2019.06.030.

[2] V.N. Hoang, G.W. Jang, Topology optimization using moving morphable bars for versatile thickness control, Comput. Methods Appl. Mech. Eng. 317 (2017) 153–173. doi:10.1016/j.cma.2016.12.004.

[3] V.-N. Hoang, N.-L. Nguyen, P. Tran, M. Qian, H. Nguyen-Xuan, Adaptive concurrent topology optimization of cellular composites for additive manufacturing, JOM. 72 (2020) 2378–2390. doi:10.1007/s11837-020-04158-9.

2. Speaker: Hung Vinh Tran

Title: On a critical Coagulation-Fragmentation equation. 

Abstract: We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of mass-conserving solutions to the Coagulation-Fragmentation equation. These solve partly a long standing open problem in the field. Joint work with Truong-Son Van (CMU).

3. Speaker: Nguyen Dang Hop

Title: Free resolutions of ordinary and symbolic powers of ideals 

Abstract: Ordinary and symbolic powers of ideals are basic objects in commutative algebra. The study of ordinary powers of ideals is motivated by the study of blowing-up in singularity theory. It has connections with multiplicity theory, the study of singularities in positive characteristic, among others. The study of symbolic powers of ideals is related to the question on the minimal number of defining equations of an algebraic set. 

One important theme on powers of ideals is the asymptotic properties of large enough powers. We will discuss recent progress on the asymptotic study of powers of ideals, focusing on invariants of their free resolutions.

4. Speaker: Bui Hai Hung

Title: How to make plans from visual input

Abstract: Optimal decision making and control are the hallmark of intelligence capability. Classic methods however only work in low-dimensional state spaces, while humans and animals seemingly can make optimal decisions from very high-dimensional visual input. In this talk, I will introduce a recent approach called "Learning Controllable Embedding". This approach uses deep neural networks to embed the sequence of visual inputs in a low-dimensional state space, while at the same time encouraging the sequence of embeddings to be amenable to classic control techniques. The result is a system that can make optimal decisions from visual input by planning entirely in this new embedding space, similar to how humans seem to make mental models of the real world.