Structural Topology Optimization Method for Morphogenesis of Dendriforms

The topology optimization method of continuum structures is adopted for the morphogenesis of dendriforms during the conceptual design phase. The topology optimization model with minimizing structural strain energy as objective and subject to structural weight constraint is established by the independent continuous mapping method (ICM) which is a popular and efficient method for the topology optimization of continuum structures. This optimization model is an optimization problem with a single constraint and can be solved by the iteration formula established based on the saddle condition. Taking the morphogenesis of a plane dendriform as an example, the influences on topologies of the dendriform are discussed for several factors such as the ratio of the reserved weight to the total weight, the stiffness and the geometry shape of the roof structure, the height of the design area, and so on. And several examples of application scenarios are presented, too. Numerical examples show that the proposed structural topology optimization method for the morphogenesis of dendriforms is feasible. It can provide diversiform topologies for the conceptual design of dendriforms.

Tree-inspired dendriforms and fractal-like branching structures in architecture： A brief historical overview

The shapes of trees are complex and fractal-like, and they have a set of physical, mechanical and biological functions. The relation between them always draws attention of human beings throughout history and, focusing on the relation between shape and structural strength, architects have designed a number of treelike structures, referred as dendriforms. The replication and adoption of the treelike patterns for constructing architectural structures have been varied in different time periods based on the existing and advanced knowledge and available technologies. This paper, by briefly discussing the biological functions and the mechanical properties of trees with regard to their shapes, overviews and investigates the chronological evolution and advancements of dendriform and arboreal structures in architec- ture referring to some important historical as well as contemporary examples.

(ρ, τ, σ)-Derivations of Dendriform Algebras

We introduce and investigate the properties of a generalization of the derivation of dendriform algebras. We specify all possible parameter values for the generalized derivations, which depend on parameters. We provide all generalized derivations for complex low-dimensional dendriform algebras.

Rota-Baxter TD Algebra and Quinquedendriform Algebra

A dendriform algebra defined by Loday has two binary operations that give a two-part splitting of the associativity in the sense that their sum is associative. Sim- ilar dendriform type algebras with three-part and four-part splitting of the associativity were later obtained. These structures can also be derived from actions of suitable linear operators, such as a Rota-Baxter operator or TD operator, on an associative algebra. Mo- tivated by finding a five-part splitting of the associativity, we consider the Rota-Baxter TD （RBTD） operator, an operator combining the Rota-Baxter operator and TD oper- ator, and coming from a recent study of Rota＇s problem concerning linear operators on associative algebras. Free RBTD algebras on rooted forests are constructed. We then introduce the concept of a quinquedendriform algebra and show that its defining relations are characterized by the action of an RBTD operator, similar to the cases of dendriform and tridendriform algebras.

Nijenhuis algebras, NS algebras, and N-dendriform algebras

In this paper, we study （associative） Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras and related algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota- Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.

Embedding of Pre-Lie Algebras into Preassociative Algebras

With the help of Rota-Baxter operators and Grobner-Shirshov bases,we prove that every pre-Lie algebra can be injectively embedded into its universal enveloping preassociative algebra.

J-dendriform algebras

In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the antieommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given.